Re: Remington 244
The .244 Remington is actually one of the older names for the 6mm Remington and the data for it can be used interchangeably. However I would sugest that you check the barrel for the rate of twist to see what bullets are the best for that particular rifle, especially since it's a rebarreled 1903 Springfield.
H322 is way too fast for this cartridge (used mostly in .223 Rem class cartridges)and is why the load only fills a portion of the case. Most loads with current powders weigh in at the 41-45g max loads with some slower powders being around 47-48+grns, and compressed to boot.
Older manuals listed H322 as an option because these other powders didn't exhist then.... One problem that can happen is a delayed ignition or secondary ignition where the flame from the primer jumps over the powder column and ignites it from the front rather than the rear. This can spike pressures and cause serious problems for you and the rifle. I would strongly suggest a diffrent powder choice for safety irregardless of accuracy.
If the rifle is capable of fine accuracy with H322, It should really shine with the proper powder for the job.
Hope this helps,