The first shipment of British Enfields (10,000 stands of arms) arrived in Savannah in late September 1861, and Enfields start showing up in the Confederate regiments in November 1861. There was a fairly steady trickle of these goods into the Confederacy through the remainder of the War, Even once the blockade was strengthened they were able to land goods in Mexico and bring them overland into Texas across the Rio Grande.
Wiley Sword's book, Firepower From Abroad is a good summary of the weapons deals with the United Kingdom and other countries. Another good source is Fred Adolphus' article in the Winter 1996 edition of In terms of overall weaponry, the Army of Northern Virginia managed to exchange the vast majority of its old smoothbore muskets for .58 caliber rifle muskets (Enfields and captured U.S. Springfields) by the late summer of 1863. The Army of Tennessee was still carrying a significant number of smoothbores in the Atlanta Campaign in the summer and fall of 1864; the Army of Trans-Mississippi was pretty much in the same shape.
Tom