I would see it this way: The obvious choice to seat 6-7 would be a minivan, and the reliable/low cost of ownership minivans are the Kia Sedona, Honda Odyssey and the Toyota Sienna. They are all decent and for a given budget should all be similarly reliable. If this is the best route to go, be opportunistic and get a good deal on one of the three.
The unconventional 6-seat options are as follows:
As has been mentioned, the Mazda 5 seats 6 over 3 rows and has some pros (cheap; available in manual transmission, good MPG, good handling) but it's also a bit cramped; persons 5 and 6 have to be a bit agile/small to fit into the 3rd row, and you don't get much space behind that 3rd row.
The Ford Taurus/Freestyle/Taurus X are wagons that you might be able to find really cheap (in a similar vein to the Caprice wagon pictured earlier, but Ford kept making them longer so you could get a newer one), and if you've got a low cost mechanic and/or if you're doing repairs yourself, these could be good options. In a similar vein, large domestic sedans used to seat 6; I believe the '11 or '12 Impala was the last of that breed. None of these are really reliable vehicles, but again, if you've really nailed down your automotive costs these cars depreciate like crazy and end up being cheap up front and used parts end up being cheap because there are so many of them out there.
And of course there are SUV's, but none of them are as economical as a minivan. Pilots and Highlanders are relatively good in the segment, but used prices are high.
There's really no "free lunch" when it comes to cars that seat more than 5.
Also note that the Matrix only seats 5.