Nothing wrong with a well done sporter, it is part of our shooting heritage.
With a Lee Enfield, it depends on which model of rifle that you have. As mentioned, stocking up has an effect on accuracy. A no.1 rifle (WWI style) has a forearm that puts upward pressure at the nosecap and an inner barrel band to dampen harmonics. Cut that away and you have a drastic change in properties.
A no.4 rifle (WWII style) had a barrel that was pretty much free floating from just forward of the receiver.
There is much more to replacing the stock than just assembling with a replacement. The trigger guard plays a BIG part in how the stock is bedding. The recoil lug must bear on the stock's 'draws' tightly. A whole treatise can be written on the subject, and has.
In your case I can only give some general advice. It is probably not the barrel, but the stocking up that is the cause for inconsistancey. For a hunting rifle, sportered stock or custom (including synthetic), glass bed it reaching just forward of the chamber or 'barrel reinforce' (the fat part) and allow the rest of the barrel to free float. Make sure that the trigger guard bears evenly on the wood below it and that the front screw is cranked in tight.
Point of impact will move as the barrel heats up, however, you will get to know how it shoots cold and it will hold zero it for the first few rounds. Good enough for hunting, unless you are are a real bad shot!
Hope this helps.