FWIW here are pages showing the various .44 caliber cartridges for target and sporting rifles from my 1903 Union Metallic Cartridge Co.
UMC Ammunition book.
1903 Ammunition Book pg 15.jpeg
1903 Ammunition Book pg 16.jpeg
It appears they offered the 2 5/8-inch Sharps only, necked case with a load of 90 grains of powder pushing a 500-grain patched bullet and a load of 105 grains of powder pushing a 520-grain patched bullet.