Maybe the new starter solenoid takes more amperage to pull than the old starter did. When hot, the resistance in the start wire may be high enough that there's not enough juice to pull the solenoid and engage the starter motor.
There's countless old cars starting up just fine with the original wiring and starters, but once in a while one needs a little tweeking to help if perform.
An advantage of using a remote relay is, instead of a 12 gauge wire supplying power to the starter solenoid you have a thick cable. The cable isn't going to lose as much power when it gets hot as the thinner wire. You could install it where the regulator was to keep it a bit hidden but you will have to run the cable to the drivers side and then from there to the starter. Having the relay close to the battery results in only a short cable to the relay and the longer cable from the relay to the starter is electrically dead until the start switch is engaged. It also re-positions the R (yellow) wire from the starter to the coil.
As stated in the link posted above, the relay is also a good place to attach accessory wires and I've used mine to hook up a remote start switch as well. Mines on the passenger fender, (see pics), doesn't look stock to those that are concerned about that but looks OK to me, and powers the Hi-Torque starter well. Isn't a Ford part, it's a type of relay that Ford also uses. Mines an ACDelco.