Tips for wannabe expats? (Singapore)

Despite me being a member here I am currently living in Los Angeles. A great city for sure, but I want to get out and a recent trip to SG has me wanting to come back on a more permanent basis. I suppose what I am trying to become is a self-made expat. I hear over and over again that the best way to move overseas is to work for a MNC and get transferred to the country of your choice. Unfortunately I work for a company that’s mostly based in the US (and a defense one at that) so there’s no hope for me to accomplish that scenario. I’ve been applying to companies there, but I feel like I’m at a serious disadvantage because I’m not in country and would need sponsorship. Have any of you gone that route, or known anyone who has? I hear recruiters are a good option. Help please?