1.) By getting to 100-120 cap using repeatable actions BEFORE reading any INT increasing books in your library (including the ones available by purchasing upgrades in the library) and attending Academy lessons, and then using the books/lessons to get higher than 120 INT. The most repeatable action to get INT to 120 (or, rather, 119) is to marry someone with INT+2 of you. You are 79 INT? Marry an 81-82 INT. Use the cafe options in the Events and Options interaction, she has a small chance to raise your INT by +1 to hers. Put her in the Academy (or if it is the Stormchild, have normal sex with her) for her to gain +1 INT so she is higher than yours again, cafe to bring your INT up by 1, repeat.
2.) Historical Society raises Academics and Science, and it has a 100 cap. If it somehow raises INT, which idk I have not checked the code recently, then it was probably already capped lower than your current INT.
3.) Ayden's RPG is a one-time INT increase.
SugarCube.State.active.variables.faction.warlord1.men = 1000