I took a look at the NEF website. I don't see any of the 35 cal. or 338 cal. family of cartridges. I am not sure where you are hunting ... but did I guess correctly that the bullet must be 35 cal. or larger ? If so that pretty much linmits you to the 40 ca. + stuff.
I agree that a 35 Wheelen would be nice, but go look for factory ammo ??? Tough to find at the "usual" outlets.
As far as accuracy ... I have firsthand shooting expeerience with only 4 Handy rifles ... 308 Win. (1), 243 Win. (2) and 30/30 Win. (1). The owners and I tried a few factory loads for each. i do not recall which ... been 6-7 years ago. My recollenction is that all 4 shot decent enough. One of the 243's was a very nice grouping rifle using a particular factory load .. I'm 95% sure that it was a Winchester Supreme load using a 100 gr. bullet ... but I do not recall any details.
I worked up a load for the guy that had the 308 Win. Came up with a nice grouping round. As I recall it was a 165 gr. Game King at about mid-velocity of a Speer handloading manual.