WASHINGTON — After George W. Bush twice carried the New Hampshire county that includes vote-rich Manchester and Nashua, President Obama turned the tables and won it in both his elections by similar margins.

Mr. Bush handily captured the Ohio county that includes Cincinnati and its mostly white suburbs in 2004, while Mr. Obama won there running away in 2012. Colorado and Virginia went for Mr. Bush, then flipped to Mr. Obama.

Now that Mr. Obama has endorsed Hillary Clinton, her advisers are eager to use his political touch in those and other battleground states and extend the Democratic streak there this fall. They see Mr. Obama as a one-of-a-kind resource — a popular sitting president — in the looming campaign to defeat Donald J. Trump.

Political strategists at the White House and in Mrs. Clinton’s campaign are just beginning to determine a specific stump-speech schedule for the president after his endorsement on Thursday of his former secretary of state. Mr. Obama and Mrs. Clinton will appear together on Wednesday for the first time since she secured the Democratic nomination, and they have chosen Green Bay, Wis., another city where Mr. Obama lifted Democratic fortunes.