Poison oak, poison ivy and poison sumac all produce urushiol, which causes an allergic dermatitis (a rash) in some people after contact with the skin. The reason that you can get a reaction even after it has been ‘on a jacket for a long time’ is that it is a pretty stable chemical.
There are two main ‘parts’ or functional groups to urushiol- a long chain hydrocarbon that is similar to cooking oil (like olive oil or sunflower oil). You know how long cooking oil can sit around without going rancid? That’s how stable that moiety (part) is. You know how, if you splatter cooking oil on your clothes, that it is a pain to remove? Urushiol does the same. That long hydrocarbon is also pretty effective at soaking into skin- it behaves like moisturizing lotion in that respect, but instead of helping…it does the opposite!
The other group in urushiol is called a ‘catechol’. It is more sensitive to degradation- particularly by oxidation. But it is still pretty stable and only very slowly degrades in air.
Because urushiol is stable, it is also persistent. A wash with 3% hydrogen peroxide (a strong oxidant- degrades the catechol) and a bit of dishwashing detergent (to remove the oil) is reasonably effective at removing urushiol.