What's the next step after a lumbar facet injection if it doesn't work?

Hard to say. Depends on your pain issue. Facet injections are helpful and can be long-lasting, however if your get short lived relief you may need to get evaluated for a radiofrequency ablation for longer lasting relief (greater than 1 year of relief is typical).
Be careful. When you say it didn't work. If you got short-term pain relief, it did work. In that case, confirmatory facet blocks and/or medial branch blocks would be the next appropriate step.
Look for another . Look for another reason for the back pain as it is not uncommon to see facet changes on imaging studies as we age. Just because there are these changes does not necessarily mean they are problematic.