Update: Actually I did find the time. It was easier than I thought, because Volokh,
in documenting the evolution of "right to bear arms" clauses, already finished my work for me. The new count is that bills of rights existed in 20 out of the 35 state constitutions when the Civil War started. On the assumption that every state constitution with a "right to bear arms" clause also has a bill of rights, that means that bills of rights were pretty common indeed even before the war. Here is the list of constitutions that contained "right to bear arms" clauses in 1860, so most probably had a bill of rights. (Note that 20 is a conservative estimate: There may well have been bills of rights without a right to bear arms clause.)
Alabama (1819), Arkansas (1836), Connecticut (1818), Florida (1838), Indiana (1816), Kansas (1859), Kentucky (1792, 1850), Maine (1819), Massachusetts (1780), Michigan (1835), Mississippi (1817), Missouri(1820), North Carolina (1776), Ohio (1802, 1851), Oregon (1857), Pennsyslvania (1776), Rhode Island (1842), Tennessee (1796), Texas (1836), Vermont (1777)