What's the most efficient to run tests?

I started working at a new company, they develop a banking software. They need to run tests to ensure the quality.

The software engineer who's far more experience than me writes unit tests for every new method, and then runs an integration tests, which included cucumber selenium and phantomjs and whatnot. Once he is satisfied that the unit tests and the integration tests run, he only then push to the master repo.

He does that everyday and made me do it, he doesn't wait for a feature to be added, he does it whenever a bug is fixed or a method is written, or anything. He argues that doing so ensures is the best, if you wait too long, you might forget about a bug. Integration tests will eventually be run on Jenkins but the procedure and time spent on them won't decrease, and unit tests will still be run by hand.

I was wondering if there's a better way of handling tests, decreasing the time spent writing them while maintaining the same code quality, both front-end and back-end.