On the AutoTempest search results page, you can change parameters and run your search again by clicking the "Modify Search" section in the menu, on the right-hand side of the page. (Or if you're on your phone, you can open the menu by clicking the "menu" button.) Or, if you prefer to get back to the homepage, you can do so at any time by clicking the AutoTempest logo at the top of the page.
If you press your browser's back button, it will take you to whatever page you previously visited. So if you ran your search from the homepage, it should take you back there. However, if you got to the results page some other way - through a link or bookmark for instance - you'll end up back at the page you were on previously.