He went into exile in May of 1936 - first to Jerusalem to prepare his case to the League of Nations, who were located in Geneva, Switzerland, then immediately to England, where he stayed for something like five years, mostly in Bath. I didn't know this before, (and I know a little about the man) but I also took a look at his wikipedia article, which claims that he also spent a lot of time in Worthing, Wimbledon, and Malvern. But that's wikipedia, so...

Anyways - he returned to Ethiopia in 1941 after Britain's North Africa campaign poured troops and resources into the region in a successful bid to extricate it from the Italians.

I know nothing about him visiting Jamaica early on and considering the crazy shit Ethiopia was going through, and the fact that a major component of Selassie's exile was to try and convince the countries of Europe to step in and use their military might to rein in Italy, I can't imagine why he would've traveled to the Caribbean. Rastafari was born in Jamaica without his direct intention, around the time of Selassie's coronation...basically he is looked at as the fulfillment of Garvey's famous quasi-prophecy - "Look to Africa, for there a black king shall be crowned."