(UPDATE: See here for more on this question)
William Shatner started thinning in his mid to late twenties. The image below from 1957's Studio One: The Defender highlights just how thin his hair was by the mere age of 28 (more on this here).
Below we have a few stills from the 1958 film The Brothers Karamazov. Here Shatner is already wearing a hairpiece, although in this case it is probably best described as a simple wig. The point though is that the hair-stylists evidently didn't have much natural hair to play around with.
Finally, below we have a TV appearance from 1960 in The Twilight Zone (episode "Nick of Time"). Here, Shatner already displays his famous lace hairpiece similar to the one he wore on Star Trek.