How I have done this is to put the faster stuff in the squares around the slower stuff. When I plant a 3x3 block for squash, the middle has the squash seeds, and radishes then carrots in two rings around that. Also, I'll pluck the middle of an early square of radishes or carrots to make room for a pepper or eggplant. But it is the nature of the beast to not be able to fill a bed early and late, but have too much to what to get in during spring...

I'm in western IN, so fairly close to your eastern IL garden. The earliest I've been able to transplant my warm season crops (beans, zucs, peppers, etc.) has been mid-May, so I don't think you need to worry about planting them in April and maybe not until June 1 with the late spring we may have this year. But yes, I do have my broccoli, cabbage, etc., planted in a separate bed in order to get the full harvest out of them without having to worry about the warm crops needing to get in there.

Other crops you might consider planting after the cool crops come out in June: basil, swiss chard, soy beans, and later sowings of bush beans, carrots & beets. Swiss chard is especially productive; books will say to plant more seeds in mid-summer for a fall crop, but I've never had to -- my spring plantings are often still going strong at Thanksgiving!

Another idea might be to start your cukes and squash inside in a big enough seedling pot that you could hold off planting them outside until mid-June. Or, of course, pulling (all or just some of) the spring crops after the initial harvest and not waiting to get side-shoots, etc.