Something called "pushback" was not needed. The Union had around 2 million men in uniform, and they all needed guns. There were almost enough Springfield rifled muzzle-loading muskets already made, to equip them, but another 436,000 Enfield rifled muskets and thousands of others still had to be imported. There was simply no way to replace all of those with breech-loaders. Even though the War Dept had experimented with breech-loading guns since John Hall's shop in Harper's Ferry in the 1820's, and knew they were the future, there was simply no manufacturing capacity , let alone the money to pay for them.

Breech-loaders were prioritized to the troops that needed them the most . It is very very awkward to reload a muzzle-loading weapon on horseback, and almost all breech-loaders went to the cavalry. Only 94,196 Spencers were purchased from 1861 to 1866, the most of any breech-loading design- there were also 80,512 Sharps, and thousands of others, purchased from a large number of arms contractors. Some infantry soldiers did purchase their own: one militia bought itself Henry rifles and gave Hood's army a very bad few hours at the Battle of Franklin. But very few were otherwise in the hands of foot soldiers.

I don't know of any specific correspondence on the subject from Lincoln to the War Dept. ( though I am sure if it exists, it's not lost) But it would be easy to predict the response: there are plenty of letters within the Dept from the 1850's saying, we will get to breech-loaders, but not right now. It would, in fact, take them until 1866 and the end of the war to begin the decisive shift away from muzzle-loading rifled muskets for all troops.

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