Any discussion about hunting rifles, and especially if you are going to be making longer shots, needs to start with what critters you are after IMO. The right caliber choice for a western US hunter that will be hunting Antelope, Wolves, Bears, Deer and maybe the occasional Elk or Sheep is going to have different needs than a hunter in, say, Newfoundland. If memory serves, your Moose are slightly small compared to, say, a Yukon moose for example?
Regardless of that, the 30-06 is generally considered to be a poor cartridge for long range hunting. Undoubtedly, you could hit a target at extended ranges (just like I can with my .270) but in order to maintain the kinds of velocities needed for good terminal ballistics, you would have to shoot a pretty small/light bullet (140gr ish).
There are no "golden rules' in shooting or long range hunting but for the most part the practitioners are shooting 'heavy for caliber', high BC bullets at velocities north of 2800 fps. There are several calibers that achieve that pretty well (assuming your gun has the right twist rate to stabilize the bullet properly). A few of the more common calibers for long range big game hunting (as opposed to varmint sized game) are 7 Rem Mag, 300 Win Mag, 300 RUM, 28 Nosler, 6.5x284. In the presence of those calibers, even the "golden boy" of recent rifle calibers - 6.5 Creedmoor - doesn't stack up very well.
At the end of the day, it doesn't really matter much if you get your bullet to the right location a country mile away, if you don't have the right velocity on the bullet to perform its job properly then you will not like the results. For my chosen caliber (7RM) my bullet will normally fall below 1600 fps around 1200 yards (depending on elevation) which represents the absolute outer limit that my equipment can perform. My goal would be to feel confident that I could live up to the limits of my equipment, even if I don't never contemplate a shot at a critter at that range.
And I'm just as happy shooting my 270 in close because its all just hunting to me.