Okay, i've gone over this a bunch, and there is a way to do what you want. It's a bit counterintuitive, but it makes sense.
This class seems designed for either a multiclass druid or any of the classes that get an 'animal companion' at effective druid level -4. Of those classes, most of them are garbage for doing damage except for one - the Mounted Fury Barbarian.
But it only gets an effective druid level of 6 at level 10, which is lame, and late, and slow, but I was resigned to it after looking at how tenaciously crappy all the other options were until I found this feat, from the Pathfinder Player Companion; Boon Companion. At the cost of a feat, it fixes up your animal companion progression, setting you up for a level 7 entry into Mammoth Rider (literally the earliest possible).
Later I also found the Mad Dog Barbarian, which gets full animal companion and BAB, but delayed rage powers (which is bad, for reasons i'll get to in a second). It's from the exact same book so.. only take it if you want the fluff.
So you take Mounted Fury Barbarian 6, then enter Mammoth Rider 1, gaining a big-ass mammoth to stomp people. Which is awesome. But then you go back and take more Barbarian levels. Why, after so much trouble? Because Mammoth Rider doesn't give much after level 1. It just makes your mammoth a bit tougher, and gives you a d12 HD and +1 BAB per level.
No, instead you want something that comes online at Barbarian 10 - Greater Beast Totem. While in a Rage, you have Pounce. Yes, that Pounce. The full-attack-on-a-charge, melee-just-became-able-to-contribute Pounce. And, with Ferocious Mount and Greater Ferocious Mount, your Mammoth now has Pounce too. Of course, this set of Rage powers (Lesser Beast Totem, Beast Totem, Greater Beast Totem, Ferocious Mount, Greater Ferocious Mount) is literally every rage power choice between 1st level and 10th, and with the level of mammoth rider, only comes online at level 11. With Mounted Combat, Power Attack, Ride-By Attack, and Spirited Charge, though, you'll
And hell, you wouldn't be playing a Mammoth Rider if you didn't want that.