The latest versions of the book only say to add 28 seconds to the 3M (or some number that would only be accurate at certain speeds). I think (definitely not sure) there used to be a standard multiplier for 3M to 5k back in the 1970s when a lot of tracks were still 440y tracks. This multiplier obviously wouldn't be 100% accurate either, but it's better than adding a flat rate. The list I'm updating, for example, uses 3M + 29.7 seconds to get an "equivalent" 5k time. Of course, this would only be accurate for runners moving at a pace around 13:53 for 3M and assumes they could keep running the same pace for the additional distance (or assumes they'd be 13:51-ish for 3M and would slow down a little, but still doesn't take into account overall average pace - a 13:00 3M runner would obviously add a different number to the end than a 14:00 runner would).
For things like high school cross country, I've always multiplied the 3M time by 1.038 to get a good approximation for a corresponding 5k time (5k is 1.0562 times farther than 3M, but runners obviously average a little slower pace when they add the extra distance). Obviously, this isn't always going to be accurate, but neither are the "accepted" conversions of using 1.08 as a factor for 1,500 to mile or 0.9942 for mile to 1,600. Yet, those are pretty universally agreed upon standards.
So is there an accepted conversion for 3M to 5k in one of those old books? If not, I'm going with 1.038.