I wouldn't nescessarily look to a school for a few hours a week, but more to a local YMCA, or dance, or art or music or gymnastics center. At this age plenty of kids are in daycare afterschool, so maybe you could drop him off at an afterschool program? Of course there is also Tball, soccer, maybe even a swimteam that he could do. In our town there is even an OT clinic that organizes childrens activities. Check the Library. I think the idea of spending some time with agemates on a regular basis is a good one, but wouldn't stress over it, as soon you will have the opportunity to spend hours per week doing extracurricular stuff.
Are there any homeschooling groups in your geographical area that you can met the other parents and see what is going on?
LOL - you can talk to a local daycare center and see if you can work something out. Overall, see where DS3 is going, and see if they can help out. A long drive doesn't sound appealing. If you stick with the idea of a private school, it may be easier to add this in September or October when the schools see what their actual enrollment is.