I'm looking for a phrase/idiom that represents when you increase your chances of winning some sort of gamble (or event with multiple possible outcomes) by saturating the field with bets. E.g. shotgunning your resume or playing the field with romantic interests.
I've always been a fan of "Hedge one's bets".
to protect yourself against making the wrong choice
And does seem to be used when literally betting on things:
Basically, hedging is just a way to reduce or eliminate the risk of a bet. You would generally look to hedge a bet when you are no longer comfortable with the bet you have made – i.e. you don’t think you have a particularly good chance of winning. The simplest example of a hedge is a bet on the other side in the game in question.
And they include this example:
Let’s say, for example, that the Yankees were playing the Red Sox, and you had bet the Yankees at -120. As the game neared, though, you became less certain that the Yankees were going to win. You could hedge that bet by betting on the Red Sox at +100, and you could do it in a number of ways. If you bet the same amount of money on the Red Sox as you bet on the Yankees then your only risk would be the juice you would have to pay if the Yankees won. If you bet less on the Red Sox than you did on the Yankees then you would be making a partial hedge bet – you would effectively be reducing the size of your bet on the Yankees. If you bet more on the Red Sox than you have on the Yankees then it’s as if you had just bet on the Red Sox.
- to be or act safe; to avoid taking a risk. You should play it safe and take your umbrella. If you have a cold or the flu, play it safe and go to bed.
(McGraw-Hill Dictionary of American Idioms)