I was having the darnedest time getting a pair of tires back on the rims and wound up using the curled end of the levers, sliding it along the rim toward me while simultaneously pushing/squeezing the tire with a thumb. this coaxed the final 8" or so of the tire back onto the rim.

I thought I finally figured out what the curled part was for only to discover it's to clip onto a spoke while REMOVING a tire. too funny actually