Sometimes you do get a second chance to make a great first impression. A "second look" visit to a residency program can be an amazing way to show just how interested you are. During a second visit, you will be more relaxed and comfortable, and the people you meet will sense that. Typically, it's a chance for the original interviewers to get to know you better in a less stressful setting. Sometimes other faculty members will be there who weren't available during your first interview. At some programs, you might even be able to join a team during rounds.
Not everyone gets invited for a second look; it's a sign that the program director liked you a lot. Remember that the interview process isn't just about you asking for a job; it's about the program trying to convince the right residents to match there. Adding an applicant to a rank list is taken seriously; it's a multi-year commitment, and the program director wants to find residents who will be great long-term additions to the team, both on and off the wards. Although a second interview visit is an added expense, you ought to think of it as an investment in your future, and accept all offers. Remember that even if you aren't invited for a second look, you can also ask to return for a second visit.
Even if you've only interviewed at one or two residency programs in total this year, the residency director doesn't need to know that. Remember, as much as you want to match anywhere, the interviewers will assume that you're evaluating multiple programs, and will try to convince you to rank that program first.
Second look visits are often a good time to ask residents about lifestyle issues (how good/bad is it really), official work hours versus unofficial expectations, would they choose this program all over again if they had the chance, what aspects of the program are their favorites, and all sorts of questions like that. You should ask where people live, how often they hang out outside of work, what kind of things they do in the area, etc. You can also ask about things like maternity leave policies, but that allows the program director to use that information when ranking you, so be very careful. Sometimes asking questions like "Do you have any relatives nearby" will let people tell stories about their own spouses and kids without you sharing information that you legally can't be asked about.
One of my clients was recently invited for a second look visit on a date when she already had a first interview scheduled somewhere else. I'll tell you the same thing that I told her: although second look visits are important, if you need to choose, you should keep your appointment for a first interview. Most of you can't afford to skip any first interviews; you want to visit as many programs as possible to increase your chances of matching. Plus, what reason would you give for rescheduling that first interview? Instead, you should see if there are any other dates available for that second look visit. You can honestly tell the program director that you want to honor the commitment you already made to another program (although you don't need to share specifically where else you are interviewing), but that coming back for a second visit is also a priority for you.
Good luck! Please contact me if I can provide any more information.