Well, let's start by saying there really isn't such a thing as a "hard mouth". Some horses do, however, become unresponsive for various reasons, normally caused by things as simple as the rider constantly pulling on his mouth while riding.
Pulling and hanging on the reins creates a horse that will lean on you and ultimately begin ignoring more subtle bit cues or even ignoring the bit (and rider) altogether. A mantra of mine during lessons, don't get into a pulling contest with your horse, you will not win. With that in mind, why would you even try?
More often that not, an owner or rider's answer to dealing with the issue is to put an ever increasingly severe bit in the horse's mouth. Unfortunately, that is never the correct answer and only serves to reinforce his aversion to the bit, often leading to a far worse problem than you had in the first place.
Sometimes an owner will bring a horse to me where the major complaint is the horse being difficult to control because he fails to respond to the bit. Your stereotypical hard mouthed horse. Since control really never is the issue, it's important to make sure the horse has the raw foundation of training and I spend a little time reinforcing those early steps. So, I go back to the basics and concentrate on a lot of lateral work, coupled with staying out of the horse's mouth.
Now, it does indeed depend on why the horse is "hard" to begin with, be it simply an issue of helping the horse understand that no one is going to pull on his mouth, a lack of training or whatever else may have caused him to shut down and refuse to respond. I want the horse to tell me how to proceed. Let's assume the horse is just shutting down and tuning his rider out.
If possible, I prefer to use a round pen because it gives me more of an opportunity to let the horse learn to carry himself instead of relying on the bit. I firmly believe in the concept of one rein stops, teaching them is important in getting the horse more soft. Also, using almost exclusively seat and leg cues along with a shifting of weight is imperative in helping the horse to learn the process of becoming "weaned" off his dependence on the bit.
With some horses, they'll 'get it' fairly quickly. Others take more time. One important factor to remember is any time you take hold of the horse's mouth, there must be a good, solid reason (as opposed to grabbing for control or picking at the horse) and you need to follow through with release so the horse begins to comprehend you're not going to be yanking or hanging on him like before.
All that said, I do expect and allow some horses (depending on the discipline) to work on the bit with contact. How can that be accomplished with a horse who's had a "hard mouth" issue? By offering release whenever possible. It's really as simple as that.