It's a done deal now, but -- is he just starting first grade (or K, depending on when his birthday falls)? If he was "having issues" in just the first few weeks of first grade or K, moving him may have made things temporarily more difficult; working with the teachers and school counselor at the original school might have helped, but maybe you tried that. Anyway it's done now and you don't want to move him again. He's possibly confused by the move -- he may have been having problems at the first school, but now after just starting to get used to that environment, he must get used to a different environment. He may even feel somewhat guilty, like "I was bad so mom and dad moved me and now I can't see dad so much and it's all my fault." On top of that he may be feeliing angry too, at you and dad and the new school as well. He may be too young still to really articulate those feelings but with the help of the school counselor (see below...) you can learn how to question him to help get those feelings to the surface.
So bear that change in mind and be certain that the teachers, principal and school counselor -- be sure to include the counselor, this is why they're there! -- are fully aware of not only the reasons you changed his school but also the fact he is now spending more time with one parent and less with the parent he'd been seeing more often. Talk with the counselor about how to talk with your son to find out what he feels -- guilt, anger, confusion -- and how to help him through those emotions, which are normal ones for a kid undergoing changes.
Both the school change and the change in which parent he sees most (not sure if you had a custody change or what) are big for a kid this age. And both are happening when he's new to school. He may be in first and not K but first is a big, big step up from K, socially and in terms of expectations.
Even if he is not acting out because of all the changes, be aware that pushing and putting hands on other kids is totally, completely a normal issue for children this age, both boys and girls, though from what I've seen, boys tend to be more prone to it. If that is his only or main issue at the new school, you are all actually doing pretty well!
See the teacher (without your son there, but preferably with your ex there, so you and he get the same information and treat this issue and any discipline in exactly the same ways at both your homes). Ask for the counselor to sit in if the teacher's OK with that. Approach it from the attitude of, "We know he's doing this and it's inappropriate. What do you do here at school to discipline for it? What specific steps and tactics do you recommend we do at home to reinforce and back up what you do here at school?" And yes, the idea of a daily or weekly report from the teacher is excellent - if the teacher has time and is receptive. You're likelier to get the teacher's cooperation if you and your ex both take that "We agree with you, it's a problem, let's help him cope better" approach, and don't defend your son. (It sounds like you won't anyway.)
If you get some form of report from the teacher, it puts your son on notice that what he does at school DOES matter when he's at home. Many kids do not realize that their parents can know what is going on in school and it's a huge revelation to them when they come home one day and mom says, "I know that today you had to stay in from recess because you and Jim could not keep your hands off each other on the way to music," and so on.
When he has not been in trouble at school, and is calm, sit down with him (after you talk with the teacher and/or counselor) and work out a way to discipline at home based on what happens at school. Not easy. But ask the teacher if she thinks a chart system -- rewards for days with no infractions, something he values, like TV time, taken away on days he has been pushing/touching and you know about it -- would work for him. A child has to be mature enough to control some impulses and connect his behavior with the consequence for this to work well. But at 6, he should be able to get that connection.
And again, this is pretty typical, unless he's really walloping kids or hitting to act out, and it doesn't sound like that.