Lawrence of Puzzlania
The Minotaur hates it when people finds clever ways to circumvent the solution he's set out. Usually, he eats them.
Once in a while, though, someone comes up with a solution that's different, not nit-picky, and, in fact, better.
This is one of those times.
Some of the visitors to our Discussion Forums cleverly observed that the problem states that no part of the desert is more than a hundred miles from civilization in any direction. Therefore, the desert doesn't have a hundred mile radius, it has a hundred mile diameter, because points just at the edge of the desert must be within 100 miles of civilization in all directions. Using this information, the following solution was devised:
Walk thirty miles in one direction. If you haven't reached civilization, you now know that civilization isn't more than a hundred miles in any direction from your new location.
Return to the oasis, resupply, and walk ten miles in the opposite direction. Drop off supplies, and return. Then head out again along the same path using the food drop to extend your range to seventy miles. Seventy and thirty equals a hundred, so you must reach civilization within that range. Maximum time to escape the desert: 7.5 days.