The .311 bullets a generally FAR more accurate in the 303 Lee Enfield than the 308 jacketed bullets as the .308 do not engage the rifling full or completely seal the bore.
There are not all that many makers of .311 jacketed bullets. For hunting the 160-180 grain are excellent for the speed this cartridge generates. If you check out Tradex they have .311 bullets from just about every one that makes them. Don't overlook the Privi and Sellier and Bellot bullets, I ahve found them excellent performers in many calibers.
In addition, many bullets made for loading the 7.62x39 are actually .311-.312 so that will give you options for the lighter weight plinking bullets your are looking for.
Hope this helps, Dean.
https://www.tradeexcanada.com/produits/87?page=4