As you put the tire around the rim, squeeze it so the beads (the open edges) go into the rim's center, which is the deepest part.

It helps to have a thin rim strip and a skinny tube that takes up minimal space inside the tire. And make sure that no part of the tube gets trapped between the beads and the rim. (This would increase the effective rim diameter and makes it even more difficult to mount the tire by hand.) Inflating the tube just enough to remove wrinkles prevents it from getting floppy and finding its way beneath the beads. But remember to deflate it before rolling the last tough section of tire onto the rim. Dusting the tube and inside of the tire with talcum powder may make a difference. The theory is that it helps the last bit of bead slide easily over the rim. (an added benefit - talc may reduce flats by preventing abrasion between the tube and tire.)