Don Stith certainly is the expert. I've just finished my second gun using his parts, and this info may be useful. I built a halfstock using a Rice 1-1/8 tapered barrel--.54 cal., 34". Whether its a halfstock or fullstock this barrel is going to make the gun too heavy for most of us to carry comfortably or shoot a lot offhand. I use mine as a cross stick gun--it groups 1/2" or better at 50 yds.
My fullstock flinter has a 1" Rice tapered barrel .54, 34". I spent Sat. sighting in this rifle and shooting it. It is has ideal weight and balance for offhand shooting--a little on the heavy side to carry all day, but hawkens weren't made to carry afoot, and I need a little weight in a .54. In my opinion the Rice barrels cannot be beat--the first three shots from the flinter made one jagged hole at 25 yards. The tapered barrel is well worth the extra money; it puts the weight between your hands when shooting offhand and makes the gun point much better than it would with a straight barrel. The .54 calibre is ideal for me--a Hawken barrel ought to be 1" at the breech or wider, I'm told, and anything less than .54 is going to make the gun too heavy for my taste. A .58 in the 1" would be lighter, and the additional recoil might be a problem for some. Having shot .62's with heavy charges, you can have my part of that unless I expect to encounter griz.
The fullstock hawken is about as pretty as a plow handle, but its lines are simple and relatively easy to get right, and its simplisity grows on you after a while.
Don's parts are first rate, by the way. Goold luck.