I am a faculty member at a land-grant, Research I university, so my perspective is limited to the way things typically work here. Still it is consistent with a number of the posts here, so it may be of some value.
Much turns here on the meaning of "turn," as other posters have noted. It is not likely that institutions will "turn" their adjuncts or visiting assistant professors into tenure-track facultoids. The department chair who goes to the dean with that request will doubltless get a puzzled, slightly condescending look.
At the same time, the degree of compartmentalization in the academic setting means that the dean's perspective is only important in certain circumstances (at least in my university). So if a position opens up, and an adjunct is a strong candidate, many departments will say "Prof. X's vita is arguably as good as anybody's, s/he has made a strong contribution to the program, and, what's more, wants to be here. We'd be crazy not to offer him/her the job, if only because we don't want to do this search again in a couple of years when someone who really wants to be in Buffalo concludes that life in Rochester just can't compare and moves on." If the faculty buys that, and this type of argument is taken quite seriously in my department, the upper administration is no longer in a position to intercede (they don't have time to micromanage the affairs of the departments).
Thus the strict terms of the "Why pay more than you have to?" arguments are only partially in play as concern adjuncts who hope to secure a tenure-track line in the department where they teach.
Much also turns on the meaning of the words, "strong candidate." Faculty members here are supposed to have a strong research agenda, without which they have little chance of tenure and promotion. So when we are evaluating dossiers we often ask each other, "Is it likely that we would be able to tenure this person?" A negative answer is a kiss of death.
What that means is that an adjunct who kept an active research agenda would be a very attractive candidate if a position opened up in the candidate's field. It is at that time that it would be a great advantage to the adjunct if we had seen the him/her close up, had a sense of how his/her teaching worked on our students, knew him/her to be collegial, etc. That's the point at which the conversation I summarized above takes place, and we may well decide to make the person an offer.
So we wouldn't hire an adjunct for a tenure-track position
- if her/his field did not match the area we were seeking, or
- if the candidate's vita didn't indicate that s/he was arguably equal to the other candidates in terms of future research.
On the other hand, if the field is right, and the research agenda is in place (standards for this differ for someone just graduating and someone who has been out for a while), adjunct status can definitely work in your favor.
I speak from personal experience as a job seeker as well. I taught at my present institution as a temporary visiting assistant professor (nine classes/year) for two years while I finished my dissertation at a third university. The year I finished the planets alligned and a job in my field come open in my present department. I applied, and at the end of a national search I got the job. It seems likely that the fact that the other faculty members knew me and my work improved my chances.