Well, it poked its head up about a hundred years ago, but then went away not long after TR finished his presidency, and didn't really resurface until after WWII.
Roosevelt was a radical departure from many of the policies of McKinley, not a continuation.
And as my inclusion of Taft above indicates, it did not end with Roosevelt. It continued onward, starting to wane with Calvin Coolidge, who kept the economic internationalism of his Republican predecessors, but started moving the party towards the anti-war position that would be prevalent for a few decades (until being shattered by the reality of the Axis), starting with the naive the 1928 Kellog-Briand Pact.
The internationalist Republican, religious and socially conservative, supportive of business-- this is what today gets labelled by paleoconservatives and liberal commentators. Yet this is not new, as the prefix neo implies. It was the norm for nearly 50 years (if not more) within the Republican party.