We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. For weak coin flipping the players have known opposite preferred values. A weak coin-flipping protocol has a bias є if neither player can force the outcome towards their preferred value with probability more than 1/2+є. While it is known that all classical protocols have є=1/2, Mochon showed in 2007 that quantumly weak coin flipping can be achieved with arbitrarily small bias (near perfect) but the former best known explicit protocol has bias 1/6 (also due to Mochon, 2005). We propose a framework to construct new explicit protocols achieving biases below 1/6. In particular, we construct explicit unitaries for protocols with bias down to 1/10. To go lower, we introduce what we call the Elliptic Monotone Align (EMA) algorithm which, together with the framework, allows us to construct protocols with arbitrarily small biases.