Any No. 1 that shoots less than 3/8" without any modifications would be left alone if it belonged to me!
I have found that most No. 1s shoot more consistently when rested near the back of the forearm, or even on the front of the receiver.
The light barreled 1A is the most likely configuration to experience problems with stringing, POI changes, etc.
A Hick's Accurizer or any fix that imitates one will usually show improvements in the above problems. You might also try a very quick, simple and cheap alternative that seems to work for a lot of people: a 1" square of truck tire innertube placed bewteen the barrel and fore-end near the tip.
