Mr. Colbert and his colleagues were hoping to capitalize on the debate’s colossal viewership and help reverse the fortunes of “The Late Show,” which after a year on the air still trails Jimmy Fallon’s “Tonight Show” on NBC and has sometimes finished third behind ABC’s “Jimmy Kimmel Live!”

The “Late Show” team was also trying to sustain the momentum from its live episodes during the Republican and Democratic National Conventions in July. In those two weeks, Mr. Colbert and his program showed a nimbleness and bite that he often exhibited on Comedy Central’s “The Colbert Report” — an improvement on his lumbering, uncertain first months on “The Late Show.”

Among other course corrections, Mr. Colbert has brought in a dedicated showrunner and he has addressed a longstanding identity crisis by reconciling with his self-important, conservative news-commentator character from “The Colbert Report,” a version of whom he has dusted off and resumed playing.

Now he has to imbue “The Late Show,” on any other night of the week when the future of the Republic is not hanging in the balance, with the same sense of immediacy that has propelled him in his live broadcasts. And he must still find the harmony between his trademark brand of pointed political commentary and the broader demands of a mainstream late-night talk show.

When he started at CBS, Mr. Colbert said in an interview last week: “People were watching me learn to play a new instrument in public. Now I really don’t care and it’s so much more fun.