If you were to walk in a straight line from the highest peak to the 2nd highest peak, at some point your head would be at a lower altitude than that of the 2nd highest peak.
There's always a counter-argument to any 'definition' applied to this pursuit - for the moment, I will play devil's advocate:
Suppose the highest peak and the second highest peak are connected by a ridge that circles around with very little drop between the two, so that in a straight line you may drop below the second peak's height, but following the ridgeline you may not.
The various elevation drops suggested are obviously arbitrary and meant to give some way to distinguish between a separate summit and a subsidiary summit. To some people, climbing the second highest peak means a peak entirely separated physically from the range that includes the first peak. To others, the second highest can be in the same range, just 'far enough away and down' to make it feel like a separate peak.
For mountainous states, both methods lead to reasonably interesting lists and goals. Your conundrum is with the flat states ... and I don't pretend to have any better answer than anyone else. If you just want to dip a few feet down between rises in the fields, then your 2nd and 3rd highest may very well be close by to the state HP. If you want to assign a required drop of 50 or 100 feet then your next highest may be far away, but may look over some higher humps that have maybe just 30 feet of prominence. The state HP is objectively defined and irrefutable (up to our trusting the USGS surveyors). There is no objective definition of the next highest points without requiring some rule - human imposed and subjective - to limn out the list.
Even so, this sounds like a good excuse to break out the topos and read away. Not a bad thing, of course!