Next year will be my third in a TT job at a relatively elite midwestern SLAC. I love the job, but I'm not crazy about the location (middle of nowhere) or where the college is going under new administration. I'd like to go back on the market to see what else is out there in the SLAC world, but I don't want to sour things with my colleagues in the department (or with the dean or provost). Can I apply to jobs without my department finding out? Should I expect a backlash for applying to jobs in my third year? Also, if I were to receive an offer, but wanted to stay, would it be appropriate to use the offer as leverage to get a higher salary or more start-up funds? Or would that be viewed as a jerk move? Thanks for your advice.
Is there anyone in your department (or another dept who knows you well) who feels as you do about the new administration? If so, could this person maybe write you a letter? If there is one person who "gets" your situation that would help. As for the location, do you have a spouse/family? If so, you can always use that as an excuse for why you are applying. If none of this applies to you then it is maybe best to keep your candidacy a secret. It's quite easy to do this at the initial stage. A lot of current professors apply without letters from their institutions for the reasons you describe and people understand this. We had one in our pool who made it to the finalists.
I think that your third year is a very common time to apply for jobs. It's a lot better than applying in year 1 or 2. You might experience backlash, but again, it is a common time to apply, so hopefully people will understand.
If you receive an offer, I'd recommend negotiating only if you are willing to leave. You school might say they can do nothing for you. That said, I think even if you are heavily leaning toward staying regardless of what they say, negotiating could be a good thing to try as long as your school is as you said "relatively elite" and in good financial standing. It will also depend on the counter-offer and whether it offers a good raise. Just be smart about it. You can say something like, "I love it here and I lean toward staying because I've had such a great three years, but this other school gave me an offer that is very difficult to refuse. I was surprised at how much I liked the department/school there on my visit." Then go on about the differences in the offer (pay, teaching load, research resources, etc) and say what your current place could do to help you make the decision to stay.
If your school is not so elite afterall or in bad financial standing, I wouldn't bother with negotiating, because they won't be able to do anything for you. And if they do give you a measly raise or research money or whatever (and you stay) and anyone else gets wind of it, the others will be annoyed given the very limited resources of the school.
I left after my third year. I was able to get a letter from someone in my department because the school was not in great financial shape so everyone understood when some junior faculty were applying elsewhere. I told a few other people in my department I was applying once I had interviews. Once I decided to leave I told everyone in my department, my dean, and my friends in other departments as quickly as possible before broadcasting the news and before the news started to spread. I did not try to negotiate, in part because I was sure of my decision, but also because I knew my school could not offer me anything. I would not have negotiated even if I had been on the fence about leaving (i.e. if I had gotten just my second choice offer but not my first choice), even though that second choice place would have been a massive pay bump and would have therefore been leverage for a negotiation. I just think that even if I had been "successful" in negotiations (like a $1K raise, way less than the $25K raise offered by the other school) it would have bred resentment with everyone else, which would have made my work environment crappy. But it was moot anyway because the first choice place came through (much less of a pay bump but a better location) and it was not a hard choice.
Going on the academic job market while already on the tenure clock is always stressful, but remember that people working outside of academia will typically hold a half dozen jobs or more during their career. Career mobility is common, so don't dread it or feel guilty. That said, how you handle things with your current employer will vary depending on the situation. If your department colleagues are supportive and trustworthy, then you might consider having one of them write a recommendation for you. Ask that they keep your search absolutely confidential. If you are unsure about how they'll respond or whether they'll keep your search quiet, you could consider seeking a recommendation from a colleague in another department who you've worked with (maybe in some significant committee or a collaborative project?).
If you let anyone at your current SLAC know you're on the market, I advise not dwelling on the negatives. Don't inform them it's because you are worried about the institution's fiscal situation or new administration. Instead, frame your leaving as a positive. For instance, when I sought a new position as a tenure track faculty member, I let colleagues know I wanted to move ahead with my career at a large institution with more research resources and wanted to be nearer to family. Both were true. I didn't focus on the various problems at my then current workplace.
As @Asstprof noted, whether you can negotiate to remain at your current place once another offer is in hand varies. If your institution is cash strapped they may not be willing to offer much. Then, outside of salary, many SLACs can't do much for you with teaching load reductions for instance or research funding. Also as @Asstprof commented, many institutions want to avoid paying one faculty member tremendously more than others, especially at a smaller non-R1 institution.
Thanks for the very helpful advice, asstprof and DocJ. Both of your points about framing the decision to leave in terms of positives, rather than negatives. These are questions I would not have been comfortable asking anyone at my current institution, so I really appreciate your time and effort responding.