Just in case anybody is still looking for it, there is an option now to just re-run the tests only:

https://learning.postman.com/docs/tests-and-scripts/write-scripts/test-scripts/#add-a-post-response-test

In the Test Results tab, you can click Refresh results in the upper right of the response pane to refresh your test results. This gives you the option to refresh your test results without re-sending the request.