"I won't let a flat tire get me down," Tom said, without despair.
Where is the pun in that?
"Despair" sounds like "the spare." Alas
"I won't let a flat tire get me down," Tom said, without "the spare".
The definition of spare at Oxford includes
A spare tire
Note: at first I thought it was a pun on "without the air", but realised it was "without the spare."
I think it's a multifaceted pun.
This is really quite a clever compound pun, it seems to me.