Ah, ok. Cram schools, I know... Been there, done that in a past life actually...
If you're talking about straight-up ESL teaching or working in a language school, then yes, they do often pay more than the MOE (which should just show the diabolical situation for potential MOE teachers even more). ESL jobs are easy to get in most of Asia, as you say, but not so much in Singapore. The cram schools that target the local market (like "I can Read", "Morris Allen", "Lorna Whiston") are slightly better again. They will often demand formal teaching certification and not just an ESL cert. This is still a joke considering the slave-driving conditions and crap pay (because $5k a month is still crap in Singapore, in real terms, for anyone who isn't single or over the age of about 25). Staff turnover is massive in these places as most teachers quickly realise they can do so much better elsewhere.
As I've said many, many times before: a formally qualified, experienced teacher should approach International Schools
for work. The pay and conditions are infinitely better. People with ESL certs only do not fit this description: they will need to stick with language schools or try their luck at cram schools. The MOE (last I heard) wants native English speakers with teaching qualifications AND preferably an ESL cert, but wants to pay less than anyone else, including language schools.