We live in Indiana. The state-wide cut-off date for Kindergarten is age 5 by Aug 1 (it changed a couple years ago). However, many school districts allow an 'early entrance' of age 5 by Sept 1. My son's birthday is the middle of July. He went to kindergarten soon after he turned 5. He's in 1st grade now and doing great. Never had any issues with kindergarten. He did go to preschool a few days a week since he was 2 so he was used to a school-type environment, being separated from us, following directions, walking in a line, waiting turns, etc. That's the biggie for kindergarten - those social skills. I don't know your son of course, but since he's already in kindergarten I'd say he'd be fine entering first grade. However - check with your new local public school to find out what they expect incoming 1st graders to know. In our school district, most kids are reading simple chapter books and writing stories daily in a journal by March or April of the kindergarten year. If your son isn't do those things in his private kindergarten but your new local public school is, he may start 1st grade 'behind' and have some catch up to do.
My son is the youngest in his class. Several kids are more than a year older than he is because apparently some parents believe that May or even April is a "summer" birthday and think 5.5 is too young for kindergarten. I think it comes down to sports.... parents are/were delaying their children just so they'll be the biggest and best at sports. The schools were I'm at are cracking down on that. The teachers really hate that 1.5 year gap between kids - it's HUGE at that young age. As my son's kindergarten teacher told me last year "if I wanted to teach 7 year olds, I'd teach 1st grade". The cutoff is there for a reason... 5 year olds, even 'young 5s' can excel at kindergarten. I have some neighbor kids who were 'delayed' a year because of a summer birthday and they all hate it now that they are in 4th and 6th grade... they are much more mature than their classmates and have trouble finding friends in their grade level.
Keep in mind that many states have a cutoff date of Oct or Nov or even Dec so they have 4 year olds starting kindergarten in Aug... and they do just fine!
Bottom line... talk with the principal, counselor and a 1st grade teacher at the new school and get their opinion. Some kid has to be the youngest in the class!