Good thoughts below. I'd try to find out what in particular, if anything, gets her to stop crying at the 10 minute mark, or does she just get tired of it? Does she seem to transition better with the teacher, the assistant, or someone else? Is there an activity that piques her interest more than others? If so, maybe you can highlight those or have that individual take over at the drop-off. That doesn't mean she doesn't like the teacher, of course - but maybe the assistant or a volunteer or the director has a better style at that moment.
Try to revisit what it is that you say and do at drop-off that makes it better (shorter) or worse (longer). Sometimes kids have crazy triggers that make no sense to us adults. I would not say "Oh I missed you while you were at school" or tell her anything fun that you did during the morning she was gone.
That said, kids just do this sometimes. They do grow out of it. If she has to finish off the year this way, it's frustrating for you, but she does actually seem to be managing, because it only lasts 10 minutes and pretty much ends like clockwork. It may not happen at all in kindergarten - new setting, and she may just turn the corner from a maturity standpoint.
That said, if you feel she's not mature enough, there's no reason she truly has to go to kindergarten next fall. Starting school almost never has anything to do with intellectual ability, but more with maturity or size or other factors. My kid was a 3-hour napper and would have had afternoon kindergarten the year he turned 5. There was no way he would have made it. So we held him a year, started him at age 6, and the rotation put him in morning kindergarten. We never thought twice about it and never regretted our decision. Talk to the preschool teachers (all who work with her) and ask the director to sit in a few times when you drop her off, and talk to the public school staff who do the screening for kindergarten. Just because you have her screened doesn't mean you're committed to starting in September. You can alway wait a year and give her another year in preschool or pre-K to get her bearings.