The most difficult thing about getting a TT position is getting your foot in the door. I do think, however, it is a misconception that you have to be “groomed” for a TT position or that you need a direct “In” from your advisor or some other mentor. I’m not suggesting that it doesn’t help to have it, but it’s not 100% necessary.
The best thing about contacts at a university that you want to apply for is that you might have a source for a little extra, off-the-record information. For instance, at my ideal university, I suspect they had some specific people in mind when they advertised their open position. Even so, I applied anyway. I did not make the cut, but at least it was good to go through the motions. Also, I was able to speak with the Search Committee Chair regarding my application package and I got some good advice on things to accent in my app package.
At the other places I applied to, I did not have an “in”. They were the random online submissions, where I had originally guessed that the Send button was just a mask for Delete because they already had someone in mind. In those cases, it is generally submit and then hope for the best.
While I know it sounds crazy but if your app package says you’re ready for a TT position, the search committee will see that. If you are ready, but your app package doesn’t say that, then they’re going to think you’re not ready. That’s why is hugely important to have spent a lot of time and effort on preparing your app package.
In my experience, most of the online submission pages do not list the search committee chair. In those cases, I do think it is good to check with the department secretary to make sure your information is received in full. If they do list the committee chair, then you should inquire with them after a reasonable amount of time (maybe 1 month) to “check the status of your application” and if there’s “any more information that they need before reviewing it”. There’s no need to elaborate further on it. Doing this once is acceptable. If they don’t respond, then either you didn’t make the cut or they haven’t reviewed the applications yet. Also, it’s very easy to brush aside an email. It’s probably better to just pick up the phone and call them. It also shows that you’re proactive. However, do not become a stalker over this. That’s a surefire way to get you un-considered for the position.
In the end, just be positive about it but also realistic. While we’d like to think we’re the special candidate, on paper there are probably 10-15 other candidates at your level that look just as good. The committee is only going to pick 3-4 for interviews. Your CV, SoT/SoR, and cover letter will get you to the 10-15 person cutoff, but it takes some luck to be picked in the top 3 after that. And once you’re in the interview, there’s your chance to shine.