Should I attempt to start making payments before my grace period ends if I start working a PSLF position during that time?

Hello! I am a 2023 graduate and expect to have a grace period until November or December for my grad loans (I recently found out that undergrad loans only receive one grace period), however, I secured a PSLF-eligible position that I start on August 1st, and I am wondering if there are any drawbacks to starting payments early.
From my understanding, payments are calculated based on the previous year's taxes. I would love to start my on my 10-year road to forgiveness asap so I am wondering if I am missing something.
Thanks so much for the help!