I am a journeyman electrician. You can use the wiring from the four prong (blade) outlet. That is the newer kind. Actually, it would be better and safer (and cheaper) to change the dryer cord to four blade. A new one can be gotten at Home Depot, Lowes, or an electrical supply store. Remove the sheet metal plate
(The dryer is unplugged) that covers the connections, and take the old cord off (don't drop nuts inside). The two outer screws on the connection block are hot, and gold colored, and the center is silver, the neutral wire. There will be a squeeze cord connector through the sheetmetal outside of the machine which needs to be on the new cord, just like on the old one. On the new cord there are 2
black wires or a red and black, they are the hots, and go on the outer, gold terminals. The white connects to the center, silver terminal. Tighten the nuts tight with the original washers in place, but snug, don't break the plastic.
There is a green wire left. If there is a green screw on the metal body of the dryer, it goes there. Otherwise you need one self tapping sheet metal screw and a hex bit, or take a drill and drill a small starter hole,
and thread it in. Back it out and put the green connector under it, it is the ground, which the code did not require before but now does, for safety, it gives a fault current another path. Put the sheet metal cover back. Turn off the breaker to the outlet, plug in the cord, turn the breaker back on, and dry.
I said to turn off the breaker on the very off chance that something got misconnected, for your safety,