>Sympathy from the armed forces and various forms of political pressure, foreign and domestic, would be far more helpful than a few deer rifles, and if you've got that your private arms aren't really necessary.
Actually, having closely studied the dissolution of a modern European country (the former Yugoslavia) and seen the fallout from up close, it's not that simple (at least not always).
What happened in that case was that although the Yugoslav government [1] retained control of most of the heavy weaponry (artillery, tanks), and although the succeeding republics (Croatia and Bosnia) were initially limited to small arms belonging to local militia and hunters, the republics did have an abundance of active and retired officers with loyalty to their ethnic homelands who were quick to raise and lead battalions of troops into the field [2].
Those troops then went on to capture some heavy arms from the Yugoslav army, and scored lots of small arms, and eventually heavy arms and aircraft, from overseas allies.
Eventually, after lots of brutal warfare, ethnic cleansing, and in Bosnia, genocide, the republics did gain their independence from Yugoslavia (with significant help from the U.S. and other countries).
What was key in this case were the experienced military leaders who quickly assumed charge in the succeeding republics.
I suspect a worst-case, nightmare U.S. scenario would look much the same, with experienced military leaders taking control of ad hoc forces in their respective states and/or regions, and then fighting with small arms until they were able to capture heavy arms, receive outside help, or were defeated/captured.
1. Essentially, Serbia and Montenegro, plus ethnic Serbs elsewhere
2. Slovenia's succession was a different story than that of Bosnia and Croatia, for a variety of geographical, cultural, and historical reasons.