Is an erratum justified for a weak result? A few years ago I had published a mathematics journal article where for one of the results I had imposed extremely strong assumptions to ensure that given conclusion is true.  Figuratively speaking, I now realise that I have used a hydraulic press to crack a peanut.   Indeed, the same conclusion can be obtained with minimal assumptions.

Needless to say, I am deeply ashamed by this sloppiness, but I am not sure if this merits publishing an erratum.  One co-author strongly opposes publishing an erratum, saying that we should simply add a remark to a follow-up paper we are writing,  while the other couldn't care less.   As the original result isn't strictly incorrect, I'm not sure what the appropriate approach is.
In my opinion, an erratum is an erratum. I mean you should use it if you really discovered a flaw in a result and you want to patch it when the patch is not evident. 

In your case, in some sense you improved your previous result and this either necessitate a new paper if the improvement is substantial or just a remark in some follow-up paper.