We present the first recoverable mutual exclusion (RME) algorithm that is simultaneously abortable, adaptive to point contention, and with sublogarithmic RMR complexity. Our algorithm has O(min(K,logWN)) RMR passage complexity and O(F+min(K,logWN)) RMR super-passage complexity, where K is the number of concurrent executing processes (point contention), W is the size (in bits) of registers, and F is the number of crashes in a super-passage. Under the standard assumption that W=Θ(logN), these bounds translate to worst-case O(logNloglogN) passage complexity and O(F+logNloglogN) super-passage complexity. Our key building blocks are: A D-process abortable RME algorithm, for D≤W, with O(1) passage complexity and O(1+F) super-passage complexity. We obtain this algorithm by using the Fetch-And-Add (FAA) primitive, unlike prior work on RME that uses Fetch-And-Store (FAS/SWAP). A generic transformation that transforms any abortable RME algorithm with passage complexity of B<W, into an abortable RME lock with passage complexity of O(min(K,B)).