Seems like to me that housing changes hands a lot right after the school year ends. People put off moving so kids can finish the semester at school then they take off and move. Just like you are thinking about doing. I am not from the Chicago area so I don't know how far, miles, it is from your current location to the area you would like to move to but is it possible to move now as the Realtor is suggesting? I drive our little one several miles one way to go to his preschool. Once he is old enough for pre-K then he will of course go to the same school as our girl who will be in 2nd in the Fall.
If I were planning on moving I would also consider moving before the school year is finished so the kids can make friends this year that will carry over Summer and into the Fall. I am not a big fan of that though. I think staying with current friends until the end sounds more fun. But I would consider it, especially if I found the most perfect house in the world and it was available right now.
Go ahead and take some time and look at what is renting. The longer you wait to move the fewer choices you'll have. I still think having a move ready to do by the end of May is the better choice though, for me it would be what I would do. I would have the Realtors in the area keeping a list of homes coming up for rent at the end of May by the middle of April. Lots of people do give notice early to their landlords who go ahead and list the homes for rent on 1st of a future month.
Good luck finding something wonderful.
Just an after thought:
If you get a chance to go looking some days in early May you can also visit the schools that the kids would be assigned to by where the houses are...you know, if you live in this neighboorhood your kids go to that school a few blocks away, the school districts usually have a website that shows a map with their boundaries. They usually also have individual schools listed and you can see each schools website and check out the teachers and their pages too. That might help you decide which areas of town you want to focus on.