The letters of recommendation for a residency is key. These people will hopefully not only write you a letter but advocate for you in the residency selection process. Fortunately, the strategy for this letter is simpler than the strategy for letters of recommendation for an internship. I really have only two guidelines:
1) All of your letters should be from someone in the specialty to which you are applying unless it violates (2).
2) At least one of your letters MUST be from where you are currently working. If you are doing an internship, all of your letters can’t be from your student days. If you are doing a specialty internship, all of your letters can’t be from your student or rotating intern days.
My recommendation is therefore as follows:
At least 1 letter from a specialist in the field at your current institution. The more the better.
If you do not have a specialist in your field at your current institution, get a recommendation from someone in a core discipline (internal medicine, surgery, emergency/critical care).
The balance of letters can be from specialists not at your current institution.
The reason you need a letter of recommendation from someone at your current institution, even if they are not in your specialty, is to demonstrate that you are not a monster. If I were to read an application from someone from a private practice internship- which did not have an anesthesiologist- and they had 4 letters of recommendation from anesthesiologists from where they went to school, I would wonder, “Did they peak in vet school? Is there NO ONE working with them now who can vouch for their medical competence? Anesthesia includes knowledge of information from all kinds of disciplines- if they can’t do basic medicine, will they be a competent anesthesiologist?”
In general, more letters from people in your specialty is good. If you have your choice of specialists, those more well-known or connected may be slightly preferable. But a great letter from just any surgeon is probably better than an OK letter from a renowned surgeon. If at all possible, those writing for you should already be boarded and have a lot of experience writing letters of recommendation. If this is not possible, unboarded people in your specialty will have to do.