Roomy and fuel-efficient family hauler

The most economical/reliable 6-seater is probably going to be a Mazda 5. It's 6 seats in 3 rows, but kind of midway in size between your Yaris and a conventional minivan. They're relatively cheap used. Beyond that, you're probably looking at a conventional minivan like a Toyota Sienna or Honda Odyssey. If you can make 5 seats work, a lot more options open up.

If even the Mazda5 seems a little large, the Honda Fit might provide the extra space you're looking for. Same size on the outside as the Yaris, but far roomier inside. And extremely reliable.

A Kia Soul might also provide the extra space you want. The 2014 and up drive much better than the first generation, and you should be able to find a 2014 within your budget.