Interesting. The parent expression is "stretch a dollar", I think, to figuratively make it bigger and therefore able to buy more (by literally buying less expensive things). The supermarket flyer writer must have had that dimly in mind when he wrote "savings to stretch your budget".
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NaturalCourtOn a supermarket circular:"savings to stretch your budget"I am not sure about the idiomatic meaning here,It means extend your budget to a sufficient limit.
NaturalCourt"The extra expenses stretched the budget.", in which "stretch" seems to mean something bad.I don't think so. There is no question of good