The Civil War Soldiers & Sailors System (CWSS) is a great site for gathering information about ancestors who served in both sides of the American Civil War.
Hamiltons on Civil War Soldiers & Sailors
The US National Parks Service website has a searchable listing of the Soldiers and Sailors that fought in both sides of the Civil War. It lists the names and unit of the individual. Currently, the Sailors part of the database is still a work in progress. Below is a list of the Hamilton Variant Surnames on the site, with the number of individuals.