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For state highpointers, once the list is finished, you might start thinking what is next. State highpointing is technically about reaching the highest point and not the highest peak. Thankfully, the next list down focuses on peaks, or you would be going back to the same places, just tagging the next boulder down. Ok, dumb joke. There is no official list of second highest points for each state. The problem is that depending on what is picked as topographic prominence, the lists could vary. A consistent prominence threshold for all states has not been found and any prominence value chosen is arbitrary. Many second lists pick a sub peak of the state highpoint causing debates.
WillhiteWeb.com

State Highpoints Second Highest

Miscellaneous
Virginia - Whitetop Mountain
whitetop
Oregon - Mt. Jefferson
Washington - Mt. Adams
Gentry Hill Mt. Adams Mt. Jefferson
Louisiana - Gentry Hill
Others
Idaho - Leatherman Peak
Examples of the problem:
The north peak of Denali has over 1,200 feet of prominence, more than enough for peakbaggers to consider it a second peak.
In Utah, South Kings Peak has 352 feet prominence, and most peakbaggers consider over 300 feet of prominence a peak in Utah.
In Washington, Mt. Rainier has Liberty Cap with 494 feet of prominence. Most peakbaggers in Washington have a 400 foot threshold.

The Peakbagger.com website has chosen to avoid the issue and go with a fun list called The Second Lap. It shows the most prominent peak in every state that is not a state highpoint. This list is much more straightforward, ignoring the cases where there are multiple candidates for the prominent peak. It still runs into its own issues and misses some extremely obvious peaks that should not be skipped over, traditional peaks well established as the second highest of a state. So, for my list below, I have skipped prominence altogether, which has tragically been the case for hundreds of years and gone with what are traditionally considered as the second highest peak in the States. This idea works for the states with mountains but I will need to incorporate some of The Second Lap list to deal with the flat states.
Grand Teton Agassiz Peak Wheeler Peak
Nevada - Wheeler Peak
Arizona - Agassiz Peak
Wyoming - Grand Teton
California - Mount Williamson
Utah - Gilbert Peak
Colorado - Mount Massive
Montana - Mount Wood
New Mexico - Truchas Peak
Texas - Bush Mountain
South Dakota - Odakota Mountain
North Dakota - Black Butte
Georgia - Rabun Bald
Tennessee - Mount Guyot
North Carolina - Mount Craig
Maine - Hamlin Peak
Vermont - Killington Peak
Connecticut - Bear Mountain
New Jersey - Sunrise Mountain
South Carolina - Hickorynut Mountain