How to raise the "Taste of Lacre" quality higher?

I am currently at 3, with 4 unusual pails of so-called snow, and I was wondering how i could raise it over 3 since I wanna unlock the option that takes 5 &quotTaste of Lacre&quot[li]

Examine it with a Semiotic Monocle gives +3 Taste of Lacre. That caps at 6 but if you use it when you are at 5 you go up to 8.