put out has 23 distinct synonyms spread across 7 of its 10 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
to cause inconvenience or discomfort to
“Sorry to trouble you, but...”
put out considerable effort
“He put out the same for seven managers”
be sexually active
“She is supposed to put out”
retire
“he was put out at third base on a long throw from left field”
administer an anesthetic drug to
“The patient must be anesthetized before the operation”
Synonyms, antonyms and the broader and narrower relations for put out come from Princeton University’s WordNet. Full sources and licences.