After that season, Alfonso joined Chievo's Giovanissimi. He was sent on loan to Pizzighettone of Serie C1 in summer 2006. He was competing with Alex Cordaz for first choice until Cordaz ended his loan.
On August 27 2007, he joined Internazionale on co-ownership deal, signing a five-year deal with the club. He replaced Fabián Carini, who left for Real Murcia, as fourth goalkeeper. He took the 71 shirt, which was previously owned by Cordaz.
On January 17,2008 Enrico made his debut against Reggina in the Coppa Italia, in the match he was forced to make two good saves in the end Inter won 3-0, and 7-1 on aggregate.
In January 2008 Alfonso was loaned out to Venezia until the end of the 2007-08 season.
On September 5 2007, during a Four Nations Tournament held at Lugano, Alfonso played the whole match of Italy under-20 national team against Switzerland. Alfonso managed to hold a clean sheet, as the match ended in a goalless draw.
On 2007-10-05 Pieluigi Casriaghi called up Enrico Alfonso, alongside with teammate Francesco Bolzoni, to form part of the Italy national under-21 football team. The national side will be involved in the 2009 UEFA European Under-21 Football Championship qualification matches agiasnt Croatia national under-21 football team and Greece national under-21 football team.
On 2007-12-03, Enrico Alfonso, alongside with teammate Francesco Bolzoni was called up to Massimo Piscedda's Italy national under-20 football team to take part in the Four Nations tournament match against Germany.
He was called up to the 2008 Toulon Tournament in France wher he played 2 games keeping one clean sheet, during the group stages, after Italy already qualified for the next round with a win.