Start / join an anesthesia interest group. Find an anesthesiologist that is known to be a good mentor by peers and start a continuous relationship with them, meeting regularly or even shadowing outside the rotation if you are able. It's hard to get an LOR from an established relationship with an anesthesiologist because you only work maybe 2 days the whole rotation with the same one.
Put powerful influential people in your corner, and invest in a relationship with them.