I've never seen any historical references of conical bullets be'n used in "original" Hawken rifles in any of the books I've read involve'n these rifles.
This tells me thet shoot'n conicals in a Hawken style rifle is a "modern idea", and is not "traditional" unless someone can prove otherwise!!
Louis Simond's, an acquaintance of Lewis Garrard's, spoke of a Cincinnatian at the Taos Insurrection of 1847 as: "Out in the pinyon, that morning with his big Saint Loui' gun - a Jake Hawkins gun, she was, eh? He had bullets an inch long, with a sharp pint - be doggoned ef they wasn't some eh?"
The history of conicals goes way back before Minie:
The English, who never succeeded in making their speculations, or practice in firearms prevail, but who always made a great noise about their improvements,
used the conical bullet for the first time in the year 1627, at the siege of La Rochelle, a Protestant stronghold, besieged and taken by Richelieu.
The false or loading muzzle, which was used to help maintain alignment while loading tight fitting conicals - at that time known as picket or sugarloaf bullets - amongst other things, was patented by Alvan Clark in 1840. His refined Wesson rifles, false muzzles, gain twist rifling, "freed" aka choke bore were used in military action as well as for target work>
"I shall forward...the amt. due Wesson for the Rifle....Everybody....is pleased with Wessons Rifle and are anxious to obtain them."
Cpt Samuel Walker, Castle Perote, Mex 1847
In 1835, match rifles began to shoot the "picket" or "sugarloaf" bullet, a conical shaped round that gave slightly better range than the round ball. It, too, used a linen patch. Because of its shape, it was difficult to get started down the barrel accurately. If the nose of that style bullet is not in alignment with the axis of the bore, one side of the base of the bullet will exit the muzzle sooner than the other, giving an erratic spin to the bullet, thus decreasing accuracy. In order to counter this, the "straight starter" was introduced.