There are a lot of different ways to describe something that is producing a strong scent or smell. Instead of using the phrase "give off odor," you could say that something is emitting a strong fragrance, releasing a powerful aroma, producing an intense or pungent scent, or even emanating a foul smell. Other synonyms for this phrase might include words like exuding, spreading, or diffusing a particular odor. Whether you are trying to describe the delicious scent of freshly baked cookies or the unpleasant odor of a dirty gym bag, there are plenty of ways to articulate the way something smells without relying on the same phrase over and over again.