Methane is a more than viable alternative and there are a few others. Of course for both we'd need infrastructure to generate it from renewable energy (as opposed to extracting using or extracting hydrogen from natural gas). In terms of energy density, methane is good enough for space flight. E.g. the spaceX raptor engines burn methane. The long term plan for SpaceX is to use renewable energy to generate methane; which as solar & wind cost drops, might actually become cheaper than using natural gas. On Mars this will actually be the only way to generate methane as there is no natural gas there.
Even just switching to natural gas would already be more sustainable as it burns a lot cleaner and there's less need for energy intensive processing of crude oil to produce kerosine (not to mention shipping oil around the globe using oil tankers & trucks powered by bunker & diesel fuel). Kerosine is nasty stuff in terms of sustainability.
So, there's a path to 0 carbon for jets (supersonic and otherwise). Even compressed natural/synthesized methane gas is viable for GA and there have been some prototype planes for that. For jets, you'd want the cryogenically cooled liquid variety because it's easier to store (lighter tanks).
Of course this would require the usual suspects to get off their ass and start innovating. E.g. Boeing had some plans for this https://www.wired.com/2012/03/boeing-freezes-design-with-liq... but not much seems to be happening. In light of Boeing's recent trouble with the 737 Max, this might be worth revisiting for them. Also the blended wing concept that Airbus demoed recently would apparently be quite good for holding liquid methane tanks. Also a 60% reduction in fuel cost sounds like it should be a nice incentive.
I wouldn't be surprised to see SpaceX pivot to producing jet engines at some point as it's an obvious thing to do and they seem to have a lot of competence around the logistics and technology involved with generating, handling, storing, and burning methane.
In any case, SpaceX is already doing supersonic flights powered by methane every time they launch a Falcon 9 and they have some wacky notion of actually using their spaceship to enable 90 minute hops to the other side of the planet.
Guilt free sustainable supersonic travel is possible but it requires people to start thinking out of the box. IMHO the first aviation company that nails this is going to do extremely well.