  In an earlier paper, the authors proved the Giroux Correspondence for tight
contact $3$-manifolds via convex Heegaard surfaces. Simultaneously, Breen,
Honda and Huang gave an all-dimensions proof of the Giroux Correspondence by
generalising convex surface theory to higher dimensions. This paper uses a key
result about relations of bypasses to complete the $3$-dimensional proof for
arbitrary (not necessarily tight) contact 3-manifolds. This presentation
features low-dimensional techniques and further clarifies the relationship
between contact manifolds and their Heegaard splittings.
