Leadership does allow for monstrous companions, many of which can potentially be used as mounts (though since nearly all are intelligent, they may not allow you to ride them for any number of reasons). All such companions take a level hit though. For example, if you were permitted to have an 8th-level companion, you could get yourself a pegasus steed, with two class levels tacked onto it.
You may also want to look into supplemental paperback books, like Animal Archives or Cohorts and Companions, that allow for additional options.
If you still want a steed, but can't take Leadership, you may want to take the Animal Ally feat instead. It will get you an animal companion, treating you as a druid of your current character level -3. You can follow up with the Boon Companion feat to erase the level penalty.
However, if your party is too large, your GM may disallow those other options anyways. I for one would not hesitate to disallow additional companions if the party was already quite large. Slowing things down with yet another companion isn't really fair to the others.
Many GMs outright ban Leadership out of hand, simply because of how powerful it is.
My advice is to work with your GM on this, not fight him on it, regardless of what the rules say.