Before we get any deeper into a "discussion" of whether a question has been answered or not, here is my thinking on what we know so far.
The problem could be either with the HPLC instrument itself, or the separation method, or the samples might be unstable in the vial.
The analyte is sensitive to moisture.
Each time that a sample is injected, the injection volume plus a bit more for loop fill or whatever is removed from the sample vial. This creates a small vacuum in the vial which can have (at least) two effects. The most commonly problematic one is that the autosampler cannot subsequently suck enough sample out of the vial to give a consistent injection volume. This usually gives eratic peak areas. The other problem, which is specific to this case, is that air gets sucked into the vial through the needle hole in the septum and the moisture in the air reacts with the analyte. Each injection introduces more moisture, and more analyte disappears - hence a progressive decrease in peak area. Critically for the present case, as long as the septum is intact the sample is stable.
To discriminate instrumental and method problems from sample instability due to air getting into the vial is straightforward - inject a series of identical samples from different vials. If the problem is due to air contact the areas will stay the same - hence my attempt to clarify whether the next day injections were from the same vial. Running a series of injections from different vials is, to my way of thinking, a lot easier and quicker than redeveloping the method, or troubleshooting the whole instrument. I would even go so far as to say that as long as the method passes QC with injections from different vials, that the sensitivity of the sample to air contact during multiple injections from a single vial is not a problem at all, because the method can de designed to avoid multiple injections.
Peter