Jose Brox has investigated and reported on a few algorithms for generating Grimm maps.  I will update this answer later with a link. One algorithm essentially works on an interval of composites and (working from greatest to smallest) assigns the largest available prime number that divides the integer. (Actually, it is a little more subtle, working only on those intervals that aren't automatically provided a map by a theorem of M. Langevin. I may expand on these subtleties later.) **Edit 2018.10.21**  Here is the promised link. Apologies for the delay!
[Link to EACA slides of Jose][1]
[1]: https://drive.google.com/file/d/1sdaxf8iz9W_hYc4sFNRtdP-G2JVTH11j/
**End Edit 2018.10.21**

I am also using this space to list some updates on the problem that I included in the Short Communication at ICM2018.  Briefly, S was run up to 10^12, with the last two failures at 5^13 with a jump of 5 and 23^7 with a jump of 23, which means no failures found in [4*10^9,10^12]. Also, computations on $f_2$ and modifications I call $g^p$ were performed: $f_2(n)$ is $O(n^{0.44})$ in the observed range and suggests $O(n^\epsilon)$ is  appropriate to conjecture as the actual order of growth, and $g^p$ appear to be the same order of growth as $f_2$ away from powers of $p$, contrary to what was hoped.  Also $f_2$ (and also $g_p$ for $n$ not below and close to a power of $p$) is much bigger than needed for an upper bound for the jump sizes made by $S$.

I am also using this to herald the companion paper, which has not made it to Arxiv yet. Some other items to be included: a mild extension to Langevin's result that an injective prime divisor map exists on certain arithmetic progressions of length n, where the extension is that the progression is allowed to contain one large divisor of lcm(1...n); a proof of the repetition of the version of S where finitely many primes are used; an explicit upper bound on f_2 after Erdos and Selfridge; and suggestions for further research.

To those who discussed this work with me in Rio, many thanks for your time and patience. I intend to reward it soon with an enjoyable account.  Thanks again to MathOverflow for allowing me to ask and answer such questions here.

Gerhard "Also Looking Forward To Write-up" Paseman, 2018.08.08.