  A k-queue layout is a special type of a linear layout, in which the linear
order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a
nested pair. The optimization goal is to determine the queue number of a graph,
i.e., the minimum value of k for which a k-queue layout is feasible. Recently,
Dujmovi\'c et al. [J. ACM, 67(4), 22:1-38, 2020] showed that the queue number
of planar graphs is at most 49, thus settling in the positive a long-standing
conjecture by Heath, Leighton and Rosenberg. To achieve this breakthrough
result, their approach involves three different techniques: (i) an algorithm to
obtain straight-line drawings of outerplanar graphs, in which the y-distance of
any two adjacent vertices is 1 or 2, (ii) an algorithm to obtain 5-queue
layouts of planar 3-trees, and (iii) a decomposition of a planar graph into
so-called tripods. In this work, we push further each of these techniques to
obtain the first non-trivial improvement on the upper bound from 49 to 42.
