As for part-time degree, you have to be careful that it will be recognised by your future employer though. Reading through the forums I see quite a bit of people have bad experience. I also had once because I did part-time bachelors before doing a full time one-year honours but still it came back to haunt me (even with my first class honours, yupp reality hurts). But if you gonna do a second degree, might as well do a masters in another field. I'm sure there any quite many fields that accept a psych degree. Of course masters courses do prefer indivs who already have some work experience prior to the course as generally have job placements.

Another position that might be interesting is user experience kind of jobs although scope depends on the company itself. So basically you help to improve designs of apps, websites etc. based on studying people's interactions with them. Its a bit of cognitive pscyh plus consumer psych as it can help with product placements etc. On this technological age I think this field will definitely bloom eventually although not sure when in sg but its already starting as I've seen some openings in sg before so not impossible.

For student care, what kind of role are you doing now? Not very familiar with this area. But I guess for managerial role its more about being to manage people than about field although an understanding of the organisation's processes will help greatly.

As for 3. I guess you could do so. just research on the positions you are interested in and which are the more specialised skill sets needed and that you lack. Although you could probably do that part-time while working in SW field (maybe even sponsored by org). I'm sure they take those with gen psych degrees. Esp if you don't mind making home visits etc.

So sorry for the uber uber long post! Hopefully, some part of this post is of use, if any =P Just rambling off the top of my head so if some stuffs are a bit patchy, please be kind.