I would think it highly depends on the school. Tulane for example ONLY wants your pre-med committee letter and if not that then your 2 sciences and I think 1 non-science letter.
I had 1 pre-med committee letter and then 2 letters from attending physicians, one a surgeon I had volunteered with for 2 years and another from a medical attending I volunteered with in another capacity for a similar time span. My pre-med "committee" letter was actually written by a single person (head of the biology, math and science department), he called a few of my professors and asked them questions about me, but they did not write any of it. He was also my pre-med advisor, so he knew me very well. He asked me to contact two or three of my professors I had good relationships with, and ask them if they would mind assisting him in writing a letter for me. I felt my "academic" letters were greatly helped by the fact that I went to a small school where all the professors knew your name and all about you.
I think most schools would have let me get away with only a pre-med committee letter, while others wanted that, plus some "additional" letters. While Tulane for example wanted nothing but the essentials.
During my interview I found that the interviewers referenced my pre-med letter but never once my additional letters. So I would bet any extraneous LOR carries much less weight than the essential letters do.
Of course some people say if you can get one from a prestigious type it could help, but I don't know. I think the best letter is from someone who really knows you well and if you have that, then sure go ahead and send it. If not don't try to snag some letter from a person you hardly know. I was very lucky in that I got three letters written and all three knew me well and they were all very good letters (from what I understandd, I never read them).
Another thing to remember is, if you are getting a letter written help your writer help you. Have your PS, AMCAS activities, transcript and other items available to send to them so their writing process will be easier and likely include the things you are proud of.