I can usually remount tires by hand, but mine are slightly fatter touring tires and I understand racing tires and rims can sometimes be extremely tight. To avoid pinching the tube, add just enough air to give it some shape and inspect it all around the rim to make sure it is inside the tire and rim before you remount the tire bead. Also the tube has a reinforced area around the base of the stem. Push the stem partway inwards and make sure that the reinforced section is inside both the tire and rim. If you have to use tire levers to finish remounting the bead, make sure the tips are rounded and smooth and that you don't catch and pinch the tube. The last part of the bead to go into the rim should be well away from the stem of the tube. Once the tire is in place, pull the stem out. Look at the sidewall of the tire all around the rim to make sure it is even, so the bead is sitting exactly where it should.