Code P0882?

There is not a procedure that needs to be done when the battery is replaced. But you might have had a voltage spike that caused problems with the TIPM which are common to go out. Please check the fuse #33 10 amp which powers the TCM. The fuse should have power all the time which is in the fuse panel under the hood.

https://www.2carpros.com/articles/how-to-check-a-car-fuse

The P0882 error code indicates a problem with the Transmission Control Module (TCM) Power Input Signal being low.

Here is the fuse location which is inside the TIPM for clarity. Check out the images (below). Please let us know what happens.

Fuse tested okay with12.01 volts on each side.

I cleaned the positive and negative cables where they connect to the firewall and the voltage at the fuse increased from 12.01 volts to 12.27 volts.
I then went through the procedure for resetting the TCM and drove the vehicle.
All appears to be functioning properly, and no trouble codes are on my scanner.
Hopefully all is resolved. I’ll let you know if it changes.
Thanks,

I left the vehicle at home for 12 hours. When I attached the scanner and turned the ignition to the “On” position, the P0882 trouble code reappeared, and the transmission was back in the “Limp mode”!
Any suggestions on what I can try next?

This means there is a problem with the TCM or one of the shift solenoids is shorted out. When it was working how did it shift normally? If so, I would try replacing the transmission solenoid/pressure switch assembly. Check out the images (below). Please upload pictures or videos in your response of any problems so we can see what to help you with. Clear the codes once completed.

It shifted just fine once I reset the TCM. I drove it on the streets and then on the freeway at high speeds. Obviously by removing the battery cables to clean them, it reset the PCM thus allowing everything to operate normally for a while with no trouble codes.
Once the vehicle set still for a while trouble code P0882 reappeared. Being this issue has never existed prior to the battery failure, do you think there could be a ground issue?

Yes, that could be, something is making it go into limp mode which causes the code. There is no TCM just a PCM, most of the time is will be the solenoid pack but it could be the TIPM is going out or a bad connection.

Here are the wiring diagrams and a guide to help you do some testing:

https://www.2carpros.com/articles/how-to-check-wiring

You might want to run a CAN scan as well to see if any codes are stored. You can get a CAN scanner (Controller Area Network) which will work on most cars from Amazon.

Here is a video to show you how:

https://youtu.be/u-4syLc-ifQ

Check out the images (below). Please let us know what happens.