Can I use my paycheck stub to get the loan process started from my new job or do I have to use the one from my last job??

What loan? You need your actual W2 to file.

The Advance Refund loan won't be available until after January 1 and just when you are ready to file.  And only if they offered it to you.  You can't self select it.  Then you don't get it until AFTER your efile is Accepted by the IRS.

https://turbotax.intuit.com/refund-advance/

It says......after IRS acceptance.

And see this FAQ

https://ttlc.intuit.com/community/refunds/help/about-refund-advance/00/27333

And…..do not try to prepare a tax return without ALL of your income documents; you must include every 2023 W-2 on the same tax return.  Employers have until the end of January to issue W-2’s.   Wait until you have all of your W-2’s.