Will an LLM degree open more doors in academia?

I hold a JD degree and have taught business law at the college level in China for six years (at US-Chinese joint ventures). However, I would like to progress in my career, but law schools in the Asian and Oceania markets increasingly want more than a JD. U-HK, for example, for their professorships, either want a JD and another graduate degree or a PhD/SJD in law. Same with law schools in New Zealand, for example, so I believe I need more training. There will be heavy competition for positions at the schools that I want to teach, so I want to gain any advantage that I can.

So, would it benefit me more to pursue an LLM or PhD/SJD, or do I need both?

I admit I don't have a lot of information about this. Also, I may have the opportunity to complete an LLM at a top school in China under a full-tuition scholarship. However, the program is in Chinese law, and I want to expand beyond mainland China and focus on international business law and a smaller focus on comparative law. Should I take the LLM in Chinese law, since it's paid, it's at a good school, and would satisfy a school like U-HK's requirement? Will having this credential stand out to hiring committees?

Just looking for a little guidance, and would appreciate any that you provide.