Need help tracking an email for my sister (incl reward)

Does she have no other way of contacting them to verify the new email address?

Translation did not work.
Do you need something like this?
Forget if irrelevant.

Is it an email alias?
[gmail]
Add or remove an email alias - G Suite Admin Help
https://support.google.com/a/answer/33327?hl=en
[hotmail]
Add or remove an email alias in Outlook.com - Outlook
https://support.office.com/en-us/article/add-or-remove-an-email-alias-in-outlook-com-459b1989-356d-40fa-a689-8f285b13f1f2

Isn't this the case?

Also, in the case of simply using Gmail or smartphones such as Android, after creating the transfer and transmission contents, selecting transmission, it waits for a while until the transmission queue is processed.
To send it quickly, you need to select the remaining tasks in the “Scheduled Folder” and select Send.

Take all this with a grain of salt, because it's quite easy to forge all the data inside an email to look as if it came from the official mail servers from the White House and passed by the Kremlin. Also, due to the nature of the data you have to examine, you can't rule out the possibility that even if things look weird, the email is 100% legit.

First step is locating another email she knows comes from Jess, the most recent one, as recent as possible. If there has happened a long time between both, then this method would not work, and she should inquiry Jess directly about a proof of identity - something they two only know, a pic, whatever. Actually, if a lot of time has passed since last contact from Jess - that's what she should do anyway, instead of trying to figure out if the email is legit.

Another method is answering the email but sending it back to the original outlook address and see if Jess receives it - if she does, 'oops it' was just an accident'. If she doesn't, insist that you sent something to the original address by mistake, and if Jess can't access it - the forward thingy was forged. I've suggested this method before at least once with good results, and it should work for you since Jess is claiming to still have access to the old email.

Ok, you have an recent email you know it's from Jess, and the new one. Open both emails and view their source code (original format). To do this, there are several methods - Exporting to .eml and opening with a text editor like Notepad - in Thunderbird it's Ctrl+U - you must have a way to view the headers and such.

On the headers, look for something like this:
X-UI-Sender-Class: 214cdeff-fd2f-45c7-a636-f5d79ae31a79
Received: from [xxx.xx.25.162] ([xxx.xx.25.162]) by web-mail.xxxxx.com
(3c-app-xxxxxxxx-lxa02.server.lan [xxx.xx.45.3]) (via HTTP); Wed, 16 Oct 2019
14:45:22 +0200

There are several Received fields showing the path of the mail, but the one under X-UI-Sender-Class should be the last one, and the first in the path. Since these headers are often done in many ways, it is possible you don't have the X-UI-Sender-Class part - in this case look at the last Received field (from the non-forwarded part). It will show the IP that the mail server saw from the sender of the email (censored with xxxx). Note that it has some extra data like 'via HTTP'. It usually means it has been sent not from an email client but from a web client.

If the IP after Received is the same in both emails, you know they have been sent from the same computer, regardless of the method they used to send the email. If it's similar, then they have been sent from the same provider in the same area. As nowadays most user IPs are semi-static/semi-dynamic, it is still possible it's the same person but for some reason there was a new IP assigned. It is also possible Jess could have done it from other device (phone, public computer, whatever). As I said earlier, it is possible to forge all this stuff so there is never full certainty, as email was never designed and it's pretty uncapable of doing any kind of authentication for this kind of forging.

You can take that IP and use any geolocation service, to see if at least it comes from the area you expect, or it comes from some shady place. You probably can expect Jess to be somewhere near, and not in Nigeria or Russia or Spain (all shady places!)

Bottom line is: if every piece of data in the email is consistent, you can't say the email is legit coming from Jess - if pieces of data in the email are inconsistent or wrong, it doesn't mean it couldn't come from Jess.

I hope that helps.
(Edited with some more caveats)

You can view the message source, the headers will be something like:

Received: from foo.com ([111.111.111.111]) by bar.com ... for <xxx@baz.com>
Received: ....
DKIM-Signature: ...
Date: ...
From: ...
To: ...
Subject: ...

bla bla

Is this person someone your sister actually knows in real life? If so, why not just call? It's harder to fake a voice you know.

Don't take it wrong but you can write here anything and it CAN be the truth but MUST NOT.

Because you are a internet stranger for me i can't proof as your intention is really a good one or if you want to find someone that don't want to be found.
divorce, stalking... whatever.

Your sentences "...who he claims to be.." implying a male Jess and later "Jess send this mail from her old account to her new one and then forwarded it to my sister." a female one.
Such little things let me look and think twice.

Why wouldn't this person be able to take a video with their phone and email it? Send it by phone? Facetime? Skype? Facebook video chat? This entire thing seems an odd request since there are a variety of much simpler ways for this person to prove they're who they say they are.

I don't understand this. Is the person someone she only knows from the internet? If this is the case, there isn't really a way to verify that they are who they say they are.

But, if it's someone that she knows or even knew from the past in real life, it's simple, just email the person and give them the details for a brand new Skype account or something that you can easily throw away. I wouldn't give them her real account info in case they get extra ways to harass her. People can't fake a video call/voice call. Problem solved.

I'd forget technology and just have your sister ask "Jess" a personal detail. Better if it's something that never was discussed by email because if Jess's email has been hacked...

Why not just send an email to her "old" outlook account asking her to verify the new gmail address? What am I missing here.

I've looked at the headers posted, what can be said is this:

• (JESSNEW@gmail.com) sent an email to (SISTER@hotmail.com)
• in this email, (JESSNEW@gmail.com) quoted a previous email which was sent from (SISTER@hotmail.com) to (JESSOLD@outlook.com)

Note the emphasis on "quoted", anybody can pretend to quote anything.
Here's an example in top-posting style of a forged email I could send you pretending to be Jess:

Hey SISTER, it's me your old buddy JESS.
I'm writing you from my new email address!
I'm quoting a previous email you sent me so that you believe me :)

Sincerely,
JESS

> -------- Original Message --------
> From: Sister <SISTER@hotmail.com>
> Sent: Monday, October 17, 2017 10:10 AM
> To: Jess <JESSOLD@outlook.com>
> Subject: hello
>
> hello JESS, its SISTER
> bye
>

So we can't conclude based on the email alone that (JESSNEW) and (JESSOLD) are the same person. It may well be the same person, only they changed their primary email address or lost access to the old one, or it may be someone pretending to be the old Jess by creating a new email address with the same name but on a different provider (gmail vs outlook)!

What (SISTER) should do is email (JESSOLD) asking them to verify the new address of (JESSNEW), of course I'm assuming the old email address was not compromised to begin with...