The wider 1/2" neighborhood tip widths are typical of the "Hill style" bows. The fact that you have built an ASL doesn't necessarily make it Hill style. So, your tip widths should be in a relationship to your width at the fades and the rate of thickness taper in your limbs. Once again, a Hill style is usually 1" to 1 1/8" at the fades and 7/16" to 1/2" at the nocks with .004 to .006 per inch in thickness taper.
So, if you dimensions match those and you are at 3/8" you are about right. All three of those dimensions need to be working together along with limb length as well.
I usually build at 1.4" at the fades and 3/8" or less at the tips with .002 taper. I have made mistakes atboth extremes in the ratios for that. 002 taper rate. One is on my bench now with 1.44 fades and 1/2" tips. It bends too much in the inner limbs and of course that leaves the inner limbs having to propel too much of the outer limbs. I'll correct it by tapering the majority of the width of the limbs in a very gentle arc to tips of just under 3/8".
Another that came out at just over 1" at the fades due to form squiggling had to be tapers as a pyramid down to tips that are a hair under 1/4" with a tip overlay that does the work of the string nocks (picture below). The bow gave up a lot of draw weight, but, to me at least, final performance is more important than draw weight.
So, the best answer to your question will be possible if you post all of the dimensions of your limbs.