The 'saddle-ring' was definitely dropped with the adoption of the model 1899 carbine and the 'long' stock', August 4, 1899.

Frank Mallory, "The Krag Rifle Story", 2nd edition, page 102, wrote: "On December 5, 1898, the Hill Shops at Springfield were given the order that 'The swivel, bar, and ring will hereafter be omitted from the stock of the U. S. Magazine Carbine, Model 1898'. This order was given too late to affect the Model 1898 carbines, however, since none with stocks were subsequently manufactured..."

Some late 'short' carbine stocks, probably made as replacement stocks (or repurposed for that task), for model 1896 and model 1898 carbines, dispensed with the inletting for the 'saddle-ring bar'. (These seem to be rare).

Note - There are also 'long' carbine replacement stocks, along with appropriate hand-guards, that were made for the model 1896 and model 1898 carbines. These stocks made the earlier carbines close to the model 1899 configuration.

FWIW: I imagine the 1899 carbine signaled a change in U. S. Cavalry equipment and the way the carbine was carried. (No longer 'Slung' across the body).

There would no longer be a need for the Cavalryman's 'sling' and the 'roller-hook' that attached to the carbine's 'saddle-ring'. I am sure dispensing with the 'swivel, bar and ring' made for a stronger stock and quieter carbine.