It is often told that working under big shots (I mean, recognized professors) in a particular research field is always beneficial on the way towards getting a faculty job in top research universities (perhaps in the US). My concern is how a big professor can help his students (PhDs, postdocs, project scientists who are badly seeking a faculty job) in addition to providing a recommendation letter? Is there anything else they could favor me on? Or, in other words, is there anything else I could ask them giving me a favor for? For instance, can I ask my professors to give a phone call to the faculty search committee (where I submitted my application) to consider my application with care? Such effort is nothing but highlighting one's application so that the search committee gives a special attention to it.
This doesn't usually work that way. When your prof gets involved on your behalf, besides writing the most stellar recommendation letter they can, they're just going to pick up the phone and call if they know someone at the place you are applying. A personal recommendation beyond the usual stellar letter from someone trustworthy that you know can go a long way towards differentiating between otherwise equally viable candidates.
One of the advantages of having a well-known professor is that they can introduce you to more colleagues (and potential collaborators!) at conferences or graduate school than a less-well-known professor can. That improves your chance that somebody on the search committee knows you or of you, or has worked with you directly or indirectly on a project.