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.

From my personal opinion, I advise you to do an LLM in the UK/USA/Canada, trying to advance to a SJD/PhD degree. If you take a look at the HKU Law Faculty professors' profiles, you can easily find that most of them have either JSD (i.e. Shitong Qiao), or JD plus PhD (i.e. Bennjamin Minhao Chen), or even LLM plus JD and JSD (i.e. Huyue Zhang). A JD cannot meet the requirement. 

An LLM from China makes little difference, considering most of its faculties are familiar with Chinese law, and many of them received their undergraduate level education in mainland China. The knowledge gap in Chinese law between you and them cannot be narrowed by an one-year LLM in China.