Many military and civil applications using underwater acoustic channels (UWACs) are paired with security services, providing, for instance, secrecy and authenticity to the communication. These mechanisms however often require periodically renewed symmetric keys among two communicating devices, namely Alice and Bob. To this end, secret keys can be agreed upon using the physicallayer communication channel, leveraging, in particular, the reciprocity and randomness of the UWAC to extract a common key that remains secret to an attacker (Eve). In this paper, we propose a novel solution for the advantagedistillation part of the secret key generation procedure. After channel probing, the advantage of distillation lets Alice and Bob extract a sequence of bits from their own (analog) measurements. We propose an asymmetric advantage-distillation protocol with two novel features: i) the quantizers used by Alice and Bob are derived maximizing the secret key capacity cooperating through a public channel and ii) Alice transmits a correction over a public authenticated side channel. Numerical results prove the effectiveness of our approach, showing that exchanging a few bits of information before information reconciliation allows extracting more secret bits from the channel.