Welcome to the forum.
There are lots of good bullets for the 25-06. Any of them would probably work well from the LRH. Just depends on the shooting distances.
Here are a few very popular ones
Sierra makes well know game killers for the 25-06.
GK 100 Gr BTSP (my favorite)
GK Pro Hunter 100 Gr SP
GK 117 Gr BTSP
GK Pro Hunter 117 Gr SP
Nosler
110 Gr Accubond
115 Gr Ballistic tip
100 Gr Ballistic tip
Berger
115 Gr VLD Hunting
Any of the above would probably work great. Use the more solid ones for closer ranges and the more fragmentable for longer ranges.
Do you already have this rifle in 25-06? If not, I would suggest looking at getting it in 6.5 Creedmoor for long range whitetail. The 6.5 shoots heavier, higher BC bullets and allows for more energy down range. The Creedmoor might add another 200 yards of maximum range and that would be about 700 yards at sea level.
One of the things that makes the 25-06 so great is that it can shoot lighter bullets pretty fast. It shoots 85 Gr bullets a little over 3500 FPS. This is great for things like long range varmints, but loses energy quickly for bigger game. The 25-06 just really lacks high BC bullets for big game at long ranges. The 6.5s on the other hand excels with high BC bullets.
Guess what I trying to say is, if you are only hunting deer at longer ranges a 6.5 might be a better choice. Hornady also makes excellent long range ammo for the Creedmoor and at a very good price. You cannot beat it really.