During the monsoon season, you may be able to get a roofer to do a fix on a small leaky spot within a few days even if your roof is still wet. But that contractor needs to come back once the roof dries out completely to make the repair permanent. Insist that the roofer credit what you paid for the first repair as part of any bill he gives you later.
Roofers are always busy once a monsoon moves through, of course. It’s good to build a relationship with someone you trust before the storm clouds blow in. They may move you to the top of their list.
If there are major holes in your roof and you have loads of water coming in, a roofer can also cover major holes with a tarp secured with nails and furring strips.
But in case of such serious damage, you also need to call your insurance company to see whether your damage is covered. Generally, storm damage is covered, but normal wear and tear on your roof is not.