Singapore is positively infested with Cyber Security Organizations, however they can be skittish, so the best idea is to stalk them from downwind till you are about 20-30 meters away and then charge at them waving a doughnut... 😉
In seriousness there are quite a few places in Singapore to go to get started, here's a few:
https://www.csa.gov.sg/careers/job-opportunities the guys are always on the lookout for people to help with this sort of thing: http://www.straitstimes.com/singapore/cyber-security-agency-to-set-up-training-academy
https://www.isc2chapter.sg/sgp/?page_id=64 Join us!
Conferences - https://www.sans.org/event/secure-singapore-2018 This one is early this year before RSA, Blackhat etc(much smaller than their US counterparts), GovWare can be fun too but that's a way off.
And if none of those are what you are after there is a meet-up for everything:
Because everywhere in Singapore is 30 minutes in a cab(apart from Tuas, which for some reason still seems to be 45 minutes away from Tuas...), meeting people is both pretty easy and important so I'd prioritize that approach above linked-in etc
Lastly @CISOScott is most correct in that you need to work out where you can help out, use that to build up islands of knowledge and then link them in useful ways. By pushing the boundaries of your current role, you can use it to provide you with experience to fuel your journey through cyber security.
Have you thought about New Zealand?
This reply is quite amusing to myself. Since I migrated from the UK 18 years ago. to New Zealand and took up residence in Auckland on Waiheke Island. Shortcomings of Wellington, central hub for Government, prone to earthquakes and related damage - most people tend to move out of Central Wellington, if they can rather than live in the capital itself. But happy to provide advice or suggestions to people thinking of moving to New Zealand.
Congrats. You will surely get a job soon
In your linkedin profile, change the status to that you are looking for job change.
Mention your skills in the order that you are strong and more confident about. See if you can find a hospital, library, school, or any small organisation where you can find some work as an intern/part time security consultant so that you can show case this in your LinkedIn profile as well as your resume.
It would be great if you can read and learn about cloud services as there is a big demand of people with such skills.