not a real papal state, but many soon-to-be-Italian politicians in 1848 really saw the chance of a Federal Italy, divided in three states, with the then-liberal Pius IX as the head of the country. this movement, known as "Neo-Guelfismo", had a moderate support by the liberals, but was staunchly fought by the Republicans and the Mazzinians.
but for a real strong Papal State, the last chances would be around 1600, maybe even before, because the Italian Wars left the Peninsula in such a horrible state that even a bigger Papal State would have been a second rate power when confronted to Spain or France