Judging from the notes at the end of his ET entry, I take it the letters posted in the entry for Assistant Surgeon Simpson reside in a place named the Mariner's Museum? Is this correct? Where would this museum be, New York?
Sorry, you're not correct even though it's a fair assumption. The reference you're citing, at the end of Simpson's entry is: Mariner' Museum (1998) Titanic, fortune & fate:, Letters, Mementos, and Personal Effects From Those Who Sailed On The Lost Ship, Simon & Schuster, New York.
This book was published by the Mariners' Museum, Newport News, Virginia, as a companion volume to their Titanic exhibition. Simpson's letter requesting transfer from the RAMC is reproduced on p29 (colour). The Mariners' Museum web site is: http://www.mariner.org/. They have some information on their holdings on-line, and instructions on how to place a research request.
However, the Simpson/RAMC letter is not in their collection, but the collection of Stanley Lehrer. I do not know how to contact Mr Lehrer. Unless others more knowledgeable have a suggestion, you could do a lot worse than contact the Museum as the first 'port of call'.
The other Simpson letter, to his mother, is reproduced (b&w) in Stephen Cameron's Titanic: Belfast's Own (Wolfhound Press, Ireland, 1998). Going by the credit, this letter is privately owned too. If you would like to take this further, Stephen Cameron can be contacted through the Ulster Titanic Society: http://www.nireland.com/uts/.