Few possibilities (some are rather silly but might fit your requirements):
The Hulk could actually be less dense than a normal human. So instead of actually adding mass, hulking up is more like inflating a balloon.
This causes several undesired side effects - for example, he will now blow away in a strong wind - so you need to address that with secondary effects. Perhaps he manipulates his internal Higgs Fields to increase the effect his mass has on the environment, giving him an effective and measurable weight of ~1 ton without actually increasing his mass.
Maybe he steals it from himself. Bruce Banner might need to eat 10,000+ categories a day, without gaining any appreciable weight. That extra mass is then consumed by future-Hulk the next time it transforms, both to bulk-up and to help power the transformation. Since the transformation goes back in time, he doesn't actually store any extra mass at that time.
As a side effect, he will gain a massive amount of weight if he's set to die in the future before he transforms again, since no future consumption is occurring.
These are a staple of a lot of superhero stories. Let's say that Banner is transformed 1% of the time. If there are at least 100 parallel universes, then the extra mass of the Hulk can exists in at least one universe at a time, transferring between universes as required.
Because this would put strict limits on when he can transform, it would be best to have a large - potentially infinite - number of universes, each temporally offset slightly. That way the Hulk Mass usage will average out and he'll be able to transform basically whenever he needs to.
Parallel Dimension 2
The "Hulk" body isn't actually a transformation. It's war chassis from another dimension, and the entire body gets stored when not in use. Long ago a super-advanced race set up these Hulk Bodies as war machines, and they transferred to them in combat. Bruce Banner never actually transforms - instead the gamma ray exposure accidentally gave him access to a specific Hulk War Body. Humans weren't designed for these bodies, and he wasn't prepared for the transfer, leading to uncontrolled trigger issues and his anger management problems.
The appearance of a transformation is actually an undesired result of an over-enthusiastic programmer of this old super-advanced race, who wanted it to look cool when he switched to his war body.