The Enfields can be accurate, but unless its already been worked on by a pro it takes quite a bit of labor to get top accuracy from a run of the mill Enfield.
The No.5 carbine can be extremely accurate if you taylor a handload to the short barrel.
Muzzle blast with most standard .303 ammo is a bit harsh.
If you want top drawer accuracy the Swedish Mausers are hard to beat. I haven't owned a 96 but I've had the pleasure of putting many rounds through a couple of these over the years. I once owned a 1895 in 7mm and it was uncannily accurate, but I can't claim that all are from that single example.
I'd intended to offer to buy that rifle back from the guy I sold it to but was told last month that he had died unexpectedly and all his stuff had already been sold off by his survivors.
The 7mm is a great classic cartridge, hard to beat even these days.