Did you actually pull the fuse and check with an ohm meter or check the
fuses in place with a volt meter on the load side?  The most likely cause is
a blown fuse.

Check the amperage rating on the fuse.  It is probably something like 10 or
15 amps, then check the draw on the compressor.  If it is at or near the
amperage rating of the fuse, it will continue to blow fuses.

 There is a thermal fuse built into the back of the lighter sockets.
Buy replacement lighter socket inserts (just the cylindrical socket
guts and screw-on shell, not the entire dashboard assembly) at the
Toyota dealer, good auto parts store, or a reputable wrecking yard,
should be in the $10 - $15 range for a new one.

 And then stop using the lighter sockets for continuous loads.
Lighter elements are meant to draw roughly 10 amps for about 20
seconds, and then pop out and turn themselves off when hot.

 DO NOT TRY TO BYPASS the thermal fuse on the lighter socket, it is a
vital safety device to prevent catching the car on fire by physically
jamming the cigarette lighter element in the On position.

 If you want to run that air compressor or any other device for
longer than about 5 minutes at 5 Amps or 1 minute at 10 Amps draw
(compressors, inverters, travel coffee pot, etc.) install an Accessory
Power Socket under the dash that doesn't have a thermal fuse.  Or use
the alligator clips accessory and go straight to the battery posts.

 This is a recording...   ;-)

      --<< Bruce >>--

You have a cigarette lighter?  I had one in my '87 Corolla, it blew
around '93.  Didn't give it a thought, and I'm a smoker.   Didn't make a
difference to the car...I traded it for my 2003 Corolla.  No cigarette
lighter, I miss having one, but I love my '03.    Don't fuss.