That's a very handsome K-38, and it deserves to be shot. Yes, pick up some match ammo and see what it can do.
To be precise, the name of the model is "K-38 Masterpiece" without the word "Target". Masterpieces were indeed used a lot for target shooting, and beginning in 1950 you could get them from the factory with options like target stocks and wide target hammers.
If you want some appropriate diamond magnas on that gun, post a Want to Buy notice in the classifieds. Any K-frame magnas with a five-digit number pressed into the back of one panel will do just fine, because all five-digit serial numbers (six if you count the K) seem to have been issued before the end of 1950. That's close enough. In reality any K-frame magnas will fit that gun pretty well, but there are subtle contour changes to magnas over time, and you stand a better chance of getting your gun close to its original appearance with stocks that number closer to the serial number of your revolver.
If you want some S&W diamond target stocks like the ones Misty mentioned, those will cost a bit more but can still be found. They probably won't be numbered, but they would make your gun look like this one, a K-38 from early 1950:
Even though this revolver has both a target hammer and target stocks on it, the factory box would not have been labeled "target masterpiece." Shooters might so describe it, but it's not a model name.
In full disclosure, those stocks are not original to the gun. I put them on. The hammer may be original equipment, but I don't know yet. I am awaiting a letter on this one.