Yes, it is still a dire fight. Firstly, any bachelor of arts or bachelor of social science degree is considered a general degree. The only specialised degrees are engineering/science or medicine. Meaning B.eng, BBM, BS. The curriculum in all three of the big universities are almost all the same. So none of them are 'ahead' of any other in terms of psych. The only difference would be in
NTU where you are directly put into psych whereas in
NUS and
SMU you take a sem or two to decide your major. But in the end, the number of pysch modules taken are almost similar (except for some
NUS grads who do not take the honours year).
SMU psych grads have a slight edge if they decide to go into the finance and business jobs but that's not what we're talking about here, we are talking about psych-related jobs.
Private university grads do push the competition. Even if they dont get govt jobs, they still get pte sector jobs, hence pushing out local grads. These local grads then try and move into govt jobs hence creating a competition there. As for overseas university grads, I'd say that they create a lot of competition when they come back as well. It even worse now. Most of the eurozone and Australia are capping their foreign workforce due to a larger right-wing push. So you will see more and more of your peers who went overseas to study coming back for a job. Last time, it wasnt so bad, overseas grads could at least get their first jobs overseas and would only come back(if they chose to do so) when they've racked up some experience. No such luck now.
As an anecdotal evidence. I have two such friends who graduated from
NUS FASS without honours this year. That would be in May-June. Both in psych. None have gotten any jobs yet. Not trying to be a fear-monger or anything. But dont just bet on your degree. Ensure that you have certain extra skills to entice your employer into hiring you.