With the caveat that you should definitely check NIH regulations yourself, my understanding is there's not a ban on supplemental salary for the same job/effort, but you can't use NIH funds to increase the stipend and you run into issues if 100% of your effort is supposed to be on the grant and you're adding a separate position with a separate effort expectation.
Recipients may supplement stipends from non-Federal funds provided the supplementation is without any additional obligation for the trainee. An organization can determine what amount of stipend supplementation, if any, will be provided according to its own formally established policies governing stipend support. These policies must be consistently applied to all individuals in a similar training status regardless of the source of funds. Federal funds may not be used for stipend supplementation unless specifically authorized under the terms of the program from which funds are derived
If you read further, it seems they allow people to take on an additional 10 hour per week position paid for services performed in some circumstances.
In practice, good luck finding anything. Institutions in higher cost of living areas especially can go above grad student stipends using their own funds, but I don't know how common this is for post docs. It's unlikely to be something you can negotiate, though, it has to be policy to give the supplement to everyone in an equivalent position.