This is the second installment in a multi-part series starting with Corwin-Dan-Cohen [arXiv:1812.05707v3]. Building on previous work by Dan-Cohen-Wewers, Dan-Cohen, and F. Brown, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of Spec Z. To do so, we develop a refined version of the algorithm of Dan-Cohen-Wewers tailored specifically to this case. We also commit ourselves fully to working with the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-1 part of the mixed Tate Galois group studied extensively by Goncharov. An application was given in Corwin-Dan-Cohen [arXiv:1812.05707v3], where we verified Kim's conjecture in an interesting new case.