Yes, it should be a skills goal to not need tools of any kind to remount a tire. The techniques mentioned above should be followed to the letter and with patience, practice, and perseverance you should succeed. I will often use gloves to protect the heels of my office-worker hands while shoving the bead over the rim. It's not the last 3" that's the hardest, it's the segment from about 8" to 3" that's the hardest. The last bit usually just snaps on, pulled over I guess by the tension of the bead that's already on.

However, if you absolutely must use a tire lever, here is the trick my wife learned at a women's racing course many years ago:
Once you have all but 5-6" or so of bead on, where it starts to get too hard for your hands to push, stand the wheel on the ground with the unmounted bead lowermost. Lean it against something, like a friend, to prevent it toppling over. Keep the tube inflated just enough to stay round (to keep it away from the bead of the rim and the tire lever, which you are going to stick into the tire now.)
With your dominant hand, ease the tire lever vertically into the gap between the tire bead and the rim bead, just far enough into the cavity of the tire to engage the rim bead, and no further. Steady the heel of your hand against the rim, to control the depth better. From your vantage point looking down into the tire you can see that the tube is not going to be pinched by the tire lever when you lever it next.
With your other hand, push the tire bead inward as far as you can so the the lever has maximum mechanical advantage against the bead. (The bead of the rim is the fulcrum.) Hold the bead against the tire lever with your thumb so it doesn't slip. Recheck the depth that the lever is stuck into the tire cavity after making these adjustments.

Carefully apply downward pressure on the handle of the tire lever. Watch as you do this to ensure that, as it presses the tire bead downward toward the ground (which is really up onto the rim bead, since we are doing this upside down, the better to see into the tire), the tip of the lever is not trapping the tube as the tip moves upward toward the rim valley.. Under direct vision you can verify that the slightly inflated tube is retracting itself out of the way and the tip of the lever is moving away from the round tube, not toward it. If the tip pops out when you lever, insert it a couple of millimetres farther next try. Don't ever plunge it deep into the tire cavity.

Continue the single stroke of the lever until the tire bead snaps on. You might need to lift the wheel off the ground to get clearance to complete the stroke. This is OK because if you've done this right up to this point, the die is cast: you can't pinch the tube even though you can no longer see into the tire cavity. You might find that the section directly under the lever gets on while segments on either side have not climbed onto the rim. If the bead has not ejected the tire lever, leave it there and use your hands to encourage the other two segments to snap over. If the lever has popped out, with any luck the bead will stay on while you push on the adjoining segments. As I mentioned above, these short sections usually go on easily -- it's the longer bit that's hard.

Because with this method you don't deflate the tube during the last push, it's unlikely that there will be a segment trapped under the tire bead. No matter. Go around the entire circumference of the tire, starting and ending where you levered the bead on, pinching the sidewalls together to make sure you can't see any tube showing outside the tire bead, just as you do with a lever-less mounting. If you do see any tube, get it relocated where it's supposed to be before inflating it. (That is a separate topic.)

Mrs. C. got pretty slick with this in her women's group. I'm not nearly as adept because I haven't had to do it often enough to get practice. But I can vouch that it works.