You could buy a standard, commonly available tool just for this
purpose (to dislodge the retainer clip), but I find that a hook tool
works wonders and is faster.  If you have ever seen the common 4 piece
pick sets from any of the major tool companies, one of those picks has
a rounded end like Captain Hook's prosthetic.  I first pry between the
panel and handle to create enough space to see the clip.  Then
position the handle so that the closed side of the clip is facing up.
Twist and install the hook tool around the clip and pull up slowly, so
that you don't hook the vinyl panel material by accident.  To
reinstall the handle, install the clip on the handle first, then press
it onto the door.  You may find it is beneficial to align the handle
as far away from your left knee area as possible with the window up.
That way, the majority of the time while driving (window up for most
people) your left knee won't hit it if it tends to wander around like
mine does.

Toyota MDT in MO