There are several factory loads for the .300H&H. I suggest you look at
www.cabelas.com and click your way to the ammunition pages. My recollection is that most of them are 180 grain, which is a shame because the .300 H&H is the only .300 Magnum (along with the .300Wby) that handles 220 grain bullets, which is nice for the heavier game on the bushveld... kudu, eland... but for North America the 180grain is fine.
As an aside, I really don't understand why manufacturers went beyond the .300 H&H... no, wait... I do understand... it was to sell more rifles! American hunters have generally been duped into believing that faster is better. Winchester made the .300 Win Mag a bit faster than the .300 H&H and voila, they sold many new rifles in the new caliber because we have to have the latest and greatest. Same thing with these ultra magnums, short magnums, and so on. The .300 H&H will kill anything that the "hotter" .300 Magnums will, and it will kill them at least as efficiently if not more so. The sole exception being shots beyond 300 yards where you actually do get some benefit from the faster bullets.