The HJ-Lagermütze (from 19.6.1936 "HJ-Sommermütze") was introduced on the 10th of February 1936 and initially there was only one cord colour: Hochrot.
I haven't found the order introducing Karmesinrot cord for HJ members yet but by the 30th of July 1937 Karmesinrot cords had definitely been introduced. This is noted in an order from that date showing that DJ members on the Jungbann and
Gebiet staff wore the HJ-Sommermütze with Karmesinrot cord if their sleeve patch (DJ rune) was also in that colour. However, a correction to that order was issued on the 8th of October 1937 because the initial order had not given any guidance on the introduction of Karmesinrot sleeve patches for DJ members on the Jungbann staff. The upshot was that the DJ sleeve patch in Karmesinrot and the HJ Sommermütze with Karmesinrot cord was only to be worn by those DJ members employed on the Gebiet and RJF staff. The same conditions must have applied to HJ members (less the rune patch of course).
The orders extant in 1943 show that the Sommermütze for HJ/DJ members and leaders up to and including Bannführer/
Gebietsführer had a Hochrot cord and that "all others" wore the cap with Karmesinrot. I'll keep looking and if I find anything more I'll add it here.