I think the instability and lack of support in a hammock would cause further pain with a back injury. In fact, I would not be able to navigate getting into or out of a hammock when my lower back injury flares. Getting into, much less driving, my Mazda Miata, (2-seater with a stick shift) when I re-injured an old injury (torn ligament from weight training) was impossible.
Rest, ice, treating the inflammation with aspirin, ibuprofen or prescription pain meds followed by heat after the initial inflammation subsides is recommended. When I lived in a gated community in Hillsboro, with mainly retirement aged neighbors, I learned that one of the elderly female residents (who had osteoporosis) fell out of her hammock and broke her neck.
Even when the back injury is simply muscle strain, often the pain is heightened by muscle spasms. Any sudden movement, such as, maintaining balance while laying in a hammock, is excruciatingly painful. Having personally experienced a painful lower back injury in my past, getting into or out of a hammock is beyond my comprehension when the injury is acute. Physicians usually recommend resting on a firm mattress for adequate support with "under the knees" support with a pillow to allow the back muscles to relax.