Based on what I have read, some of the cops who actually knew Tippit well and had worked alongside him said that Tippit had a bad habit of looking away from suspects that he approached instead of maintaining his eyes on them. This bad habit is what they believed got him killed.
Add to that bad habit the fact that LHO’s brother Robert wrote in his book “Lee” that LHO had very quick reflexes. Robert based his assessment of LHO’s reflex actions on games he and LHO played on each other as they were growing up. Going from my memory only, the games went something like faking a surprise punch at each other to basically “test” the other’s reaction time.
So, it seems reasonable to me that LHO probably just took advantage of the situation when Tippit looked away and very quickly drew his pistol and began shooting. I do believe I read that LHO did also train with a pistol while in the marines. If so, I believe that that training would have also been helpful in giving LHO an advantage in this situation.