I've noticed that the way I set up instruments (ultra precise intonation up to the highest notes of each string) makes the harmonics jump very very easily, so set up is part of it as well as construction and the type of fingerboard (the harder the more harmonics you seem to get).
A neck through bass is supposed to have more harmonics, but I had the opportunity to compare two Leduc Masterpiece basses very close in construction, except the neck "fitting" : the set neck had more harmonics than the neck through.
Funnily, my bass having the most harmonics is my Le Fay Remington Steele 6 RHT CC CAP Big Block (what a name, I know), which has a bolt-on Padauk neck with a stainless steel fingerboard (fretless) and a cherry body with a crazy cherry (CC) very thick top.
The construction of this Le Fay is interesting too as it's a headless with a headstock and strings posts with a bigger body than usual.
All my main basses are sixers around 4 kilos (a bit more or a bit less), so construction or craftsmanship is also definitely another parameter to take into account as they are lightweight.