We study the phase transition of the random degree constrained process (RDCP), a time-evolving random graph model introduced by Ruciński and Wormald that generalizes the random $d$-process to the non-regular setting: each vertex of the complete graph $K_n$ has its pre-assigned degree constraint (i.e., a number from the set $\{2,\dots,Δ\}$), we attempt to add the edges one-by-one in a uniform random order, but a new edge is added only if it does not violate the degree constraints at its end-vertices. Warnke and Wormald identified the critical time of the RDCP when the giant component emerges as $n \to \infty$. Ráth, Szőke and Warnke identified the local weak limit of the RDCP and gave an alternative characterization of the critical time in terms of the principal eigenvalue of the branching operator of the multi-type branching process that arises as the local limit object. In the current paper we use this spectral characterization to study the critical time of the RDCP in the almost 2-regular case, i.e., when the degree constraint of most of the vertices is equal to 2. In this case the giant component emerges quite late, and our main result provides the precise asymptotics of the critical time as the model approaches 2-regularity. Interestingly, our formula asymptotically matches the well-known Molloy-Reed formula, despite the fact that Molloy, Surya and Warnke proved that the final graph of the RDCP is not contiguous to the configuration model with the same degree sequence.