How old are your kids? We have some preschools that go up to age 4 that are just a few days a week, or just half days. And we have one kindergarten that you can choose to send your child half-day or full day (although I have heard that most people choose to send their child for a full day, so if you want your child to only go a half day, they might have a hard time leaving if their friends are staying).
Then we have two free or democratic-type schools, like the Sudbury model, which you can send your child for any amount of time you want, whether it's an hour a week, two hours a day, one or two days a week, etc. In that case, you would function officially as a homeschooler, and would declare to the school board homeschooling. Although if the structure type stuff is what you like about school, this may not be your type of setting.
I don't know how complete a list this is, but you could try
this link.
Hope you find what you need. I have sometimes felt the same way. I love homeschooling my kids, but sometimes I go through phases where I feel like they want/need something like that. I have addressed that by simply looking for short-term classes or workshops on topics they are interested in. This has worked really great for us. My older son especially has loved taking classes like this. He's done robotics, architecture, art, magic, and more. There are always a ton of them in the summer, but even during the school year there are offerings.
Check your local community college (ours offers summer week-long classes for kids), children's museum, community center, local colleges. All of those offer classes for kids as young as 5 or 6. I used to joke that my child liked to go to school in the summer, because that's when a lot of them are. But he does them during the year too. He has also done things like a weekly math club, taught by a friend who is a math teacher. This has worked well for us, and met those needs that you are describing, but in a way that is still in line with my desire to follow the child's lead and interests. I have never made him take a class, and would allow him to quit if he started and decided he really hated it.
Also, we don't do this all the time. It's not like I am running him around to classes all the time. Sometimes in the summer they are just a week-long thing. Other times during the school year they might be once a week for 6 weeks, for example. Another choice is to get involved in a weekly co-op. Some of them are quite structured, so it's like one day a week of school. Others are more just co-ops of parents working together to provide fun and interesting learning opportunities for your kids. We were involved in one of these for almost 8 years until it dissolved.
Good luck!
