Do you "put on" a spare tire?

I think this is really "put" rather than "put on" which means something totally different.

Definition of put from dictionary.com http://www.dictionary.com/browse/put

put verb (used with object), put, put·ting to move or place (anything) so as to get it into or out of a specific location or position:

to put a book on the shelf.

You could say:

I replaced the flat tire with the spare tire.

Or more colloquially,

I swapped in the spare tire for the flat tire.

"Put on" is perfectly acceptable and very commonly used. I wouldn't feel the need to search for a substitute phrase, although "replaced" would work.

The colloquial use of "put on a spare tire" means gaining weight around the midsection (hips).

"Put on" is used in a variety of ways. You can put on a spare tire, put on a show, put on a pair of shoes or put on a coat of paint.