With Montessori it depends on the school. I've definitely heard of children starting the year before kindergarten. Of course, they would say that you should stay the two years through kindergarten in order to make it worthwhile and that probably would be correct especially with a child starting later.

What is your child in now?? I guess I don't really understand this question or maybe it is because we have always been at a daycare/preschool. At 18 months the kids start to follow the High Reach program. The kids were always grouped by age and the public school cut off dates. Example: Our child has a nov 1 birthday. Child is in the pre-K class(last class) at our pre school this year and will be in public Kindergarten next year. I've also never heard of a Montessori not excepting any child at any time as long as they had a space for them. For a pre-k class.
If you are thinking about signing up for a private Kindergarten make sure you ask your Public school system about entering into 1st grade (when your child should technically be in kindergarten). I've "heard" they will still make the child test in but have no accurate answer.

would a montessori program allow a kid in for just one year, though? I was under the impression that most want the kids in there for a full 3 years.

Another option is to try a totally different preschool like a Waldorf, a Forest School program, Montessori, etc. if you want to wait. Instead of repeating the same things, your child would be learning a different way and learning new things. I once went to a talk and the instructor said that if you were going to hold back your child before starting kindergarten, be sure to make it a productive year and focus on the things that your child needs improvement on.

My son's preschool had a JK program designed for late birthday kids. The private where he went to K also had a JK program, so we could have done that, as an alternative.

our full-day preschool has a pre-K program. We're going to do that. my daughter misses the deadline by only 3 weeks, so we may try to get her into private Kindergarten somewhere if she's ready, and then see if she can get into public 1st grade the next year.

Andrew Chapel in Vienna has a wonderful 5 day preK program with mostly summer to November birthdays.

My son has an October birthday and did a pre-K class at his preschool before K.

Most children can't get into a typical 4's program with a late birthday. There are plenty of junior kindergarten programs around. This is what we did.

Where did your child go between 4's program and public kindergarten?