
I had the same worry when I upgraded to the OLED and kept my old hard shell case. The fit was snug but fine, I just shifted the strap so it did not press the sticks and added a thin microfiber cloth on top to stop any rub, so I did not need a new case.
That approach works for most cases with the OLED too. i would just close the case and gently press around the lid to see if anything is contacting the sticks or buttons, especially if your case has a game card flap or mesh pocket that can sag. If you feel pressure, shift the strap like you did or remove bulky items in the pocket, and consider taking off thumb caps before packing. Also make sure the zipper track is not touching the edges since the OLED is a hair wider.