Small spare tire that will fit 2010 miata

Just buy a rim and tire setup that is the same size as the stock tires. That is absolutely the best way to go because you can still use the vehicle on the spare until the regular tire gets repaired or replaced.

Also, there is no chance of damaging your drive train by inadvertently having two differently sized tires on the front.

The miata is a rear wheel drive car so the different sizes should be at the front. I have Sucessfully used an rx8 spare on my 2010 miata. It just barely fits on the trunk. The flat will not fit on the trunk so either put it in the passenger seat or bungee the trunk lid.

And to add to what William said. Carry a large outdoor garbage bag folded nicely to put your dirty flat in. That way you won't dirty your interior if you put it in the front seat. I'm putting run flat tires on my 2015 Miata with 2800 miles on her. Rather be safe than sorry. They should have came on the car from the factory...

FYI the OEM MX5 wheel mentioned by the OP will not fit in the trunk so most of the comments are wrong or do not apply