Not really an answer to your question, but a possible alternative - why not frozen garlic?

I use a lot of fresh, but there are 2 different sorts of frozen I use too - one is Indian in origin & comes as a bag of 'cubes' of frozen paste. It loses some of the punch of fresh, but you can simply add more if required.
UK pricing maybe £1.10 for 500g.

The other type is 'european' & is finely chopped & loose in a bag. This is more expensive but still retains the full flavour & aroma.
It doesn't cook down quite as well as the cubes. Even though the aroma is better for a short cook, if I'm doing a long cook where the garlic should 'vanish' I will use the cube-paste instead.
UK pricing closer to £1.50 for 100g.

Either is close enough that no-one could tell once it's cooked into a dish.
Effort required == zero in both cases.

I found a fairly random supermarket link with both alternatives - Tesco UK

Re comments on the question...
Both of these are significantly closer to fresh than anything you can buy in a jar.

The extra pivot does nothing to ease the hand force required. Coarsely chopping first can help, but not by a huge amount.

I've seen (never used) something that should help a lot: a screw garlic press, either cheap or a bit more expensive (links are to arbitrary examples on eBay and Amazon). You'd want to make sure that the hand that's holding it (as opposed to the one turning the screw) has something to get hold of.

Another tool I have used is a mandolin grater (one side slices, the other grates). Mine did a good job of finely grating garlic, but did tend to waste a ragged slice of every clove.

The problem with extending the handles is that it becomes a two-handed operation because they start too far apart for one hand to close round both handles. You may also find the weak point in the press, as at this stage you're using arm muscles rather than hand muscles (I've snapped a metal garlic press using two hands).

A microplane does an excellent job with garlic and might make a good substitute for a garlic press. Whether it's easier or not would depend on the nature of the hand injury, but microplanes use arm motion rather than squeezing.

Does it have to be a press? Can the people with hand problems grip a hammer? Use a meat tenderizer and smash the garlic. It doesn't take too much strength to hold onto the handle and swing. You can even change the handle to suit the individual's need.

There is also freeze dried garlic that is pre-minced. This is an interesting option as you don't need to refrigerate it and it works very much like fresh garlic. But, it is not quite as potent and a bit expensive.

Another alternative is a garlic press that you rock instead of squeeze. I disliked the cleaning process on the normal squeeze press (hard to get into those corners inside the press), so I bought one and really like it. Super easy to clean and came with a peeler that you roll. I would think this would not aggravate hand pain related to squeezing and you can use two hands and your body weight/arm strength.

For most applications, I can usually get away with using a miniature food processor, similar to this. As it's electric, no force is required.

It usually helps if you can chop it with some other ingredients or a quantity of oil/water, as small volumes of garlic can lead to inconsistent chopping.

I cannot give a first hand review of this device, so do not promise it is more than a gimmick, but there is a product called a "Garlic Zoom" and some other names such by companies such as chef'n. Not endorsing one name or company, only giving that one because it can be used for a generic search. The basic design it a series of rotating blades in a chamber. You put in the garlic clove, close it and then roll it on wheels across the counter. Blades are gear attached to the wheels and move back and forth to chop the garlic. Worth a look.

PS. There seems to be a number of Youtube videos out there of these in action. I am not overly fond of single application items, but this may be usable for some other fresh herbs as well and is relatively small so would not waste a lot of space. No more than the press it replaces would.

OXO brand tools

The OXO company specializes in tools made easy-to-handle and easy-to-use.

Their clever designs make their products pleasant to use both by people with or without limitations in their manual dexterity. And the products manage to be elegant and attractive in appearance while also being useful.

They offer a range of garlic and herb tools.

Any other designs which requires less force?

I can personally vouch for their Garlic Press which is as easy to clean as it is to press.

_{(source: oxo.com)}

They also offer a Garlic Slicer that I’ve not yet tried.

There are a couple of options besides garlic presses. I rarely use mine anymore.
1. Use the flat of the knife to crush it really well, then chop.
2. Depending on the recipe, saute it whole then it crushes extremely easily with a fork.

There are two things to look for in a garlic press if you want it to be easy: longer handles and narrower crushing piston (I don't know what else to call it). The longer the handles, the greater the leverage on the garlic. The narrower the crushing piston, the more concentrated the force is. To be clear, when I say narrow, I mean the amount of piston that touches the garlic is smaller.

I used to have a great garlic press from IKEA that had a very narrow piston and it was much easier to use than any other garlic press I've ever had.

I've never used this kind of device, but had seen it recently and it caught my eye. It is a rocker style garlic mincer. It would allow one to use two hands and lean on the counter, which I think could provide extra leverage:

One brand of the product: https://www.josephjoseph.com/en-us/rocker

Demo Video of this brand: https://www.youtube.com/watch?v=PxK5Aderzgw

Other brands are also available.

Not an ad, I have nothing to do with them, and this might not apply to your particular hand problems. But I have found the OXO brand of kitchen tools to be in fact kinder to hands than others. They tend to have fatter, longer and softer handles, and to require less force to operate in general.

That said, for garlic I personally have switched to @Kevin's solution, I squash it with the flat of a blade and finish it off with a bit of mincing if necessary. That saves the peeling step too, which is much easier when the clove is squashed.

Tried quite a few, and the one I'm finally happy with is this one

the garlic press from Ikea. Not sure exactly why, but not only the result is fantastic, but the force needed is less than with the others I tried.

Maybe the reason lies in the shape, the metal quality, and the fact that the internal part is removable?