ON THE BOUNCE has several meanings, some of which are obvious (as something rebounds/bounces – ball, stock price, etc.) and some which are not, with one of the ‘not’ group being the answer to your question.
ON THE BOUNCE is defined in the Oxford Dictionary of Idioms, which, incidentally, is only one of two sources in which I found it listed, and the only one which has the meaning you are interested in. It is surprising that an expression which has been so widely used – I found droves of examples – has appeared in so few slang and idiom dictionaries. But I suppose that that can be chalked up to the fact that it has only been in use for a little over a decade, and not at all in the U.S.:
ON THE BOUNCE informal [[U.K. and Commonwealth]]: In quick succession (Oxford Dictionary of Idioms)
