Start with good words - You are referring to Lee Enfields, I think. American guy, Lee, designed the action; British arsenal at Enfield Locks designed the rifling profile. So Lee Enfield No. 1, No. 4, No. 5, etc. "Enfield" also refers to the P13, P14 and M1917 "Enfield" rifles, also designed at Enfield Locks (or at least the first two were, I think?) I have receivers here from all three. Long after WWI, the P14 was designated as No. 3 rifle in British arsenal stores.
I will be corrected, no doubt, by owners of these arms on this site, but there was an enterprise named "Speed". They produced hunting rifles. Using a Lee action, and I do not know what the "speed" contribution was - Lee Speed rifles. Appear to be built on early Lee action designs, maybe prior to the Lee Enfield SMLE No. 1. They have a distinctive "perch belly" shaped magazine - never newly made, always altered from standard / military design. Reputed to be the most slickest bolt action rifles every made.
I have had the privilege of working with Lee Enfield No. 1's that are so slick I have had to open the bolt to ensure a cartridge actually did chamber - I understand that was "expected" on a Lee Speed rifle.
So if you are looking for "quality gunsmith who turned out great hunting rifle" - you are probably looking for a Lee Speed.