Try not to measure your grip using a ruler, but the left hand index finger technique. Grab your racquet with the eastern forehand grip and try to fit your left hand index finger between the tips of your fingers and the palm. If it fits snugly, then you have the right grip. If it wobbles inside is too big, and if it doesn't fit is too small.
I measured my hand and I'm around 4 1/2, but with the overgrip, I find the 3/8 suits me much better. If you measured 5/8, then maybe the 3/8 will be too small for you, even with overgrips.
I remember I once got a great racquet deal but with a much smaller grip and I removed the grip completely, attached some tape to the bare grip, then reattached the original grip over it. The end result was a racquet with my grip size, but the bevels where almost gone, like holding a pipe. Not very comfortable if you ask me.
Yeah I recommend using that technique as well.
I also don't recommend buying a racket not your grip size, for various reasons.