One survivor said they made calls to French and British police asking for help but were told to call the other. Twenty-seven people died.
Interior Minister Gerald Darmanin said justice officials and staff from the finance and foreign affairs ministries will join the agency to bolster it.
British and French authorities have called it the "biggest tragedy" involving immigrants in the crossing.
The influx of people making the journey has seen several migrants injured and even killed, while increasingly frequent clashes add to the disarray.
British Prime Minister Boris Johnson said stopping the English Channel crossings depends largely on the French authorities.
"It was much more of a physical challenge than I thought it was going to be. It actually was 50 percent physical and 50 percent technology," Rob Wylie said.
Emiliano Sala was flying to Cardiff when plane carrying him and another person went missing.
The Russian Ministry of Defense said it was "amazed" by London's response.
Captain Webb celebrates the old-school bravery of the first man to swim the Channel.
The decision to prosecute 40-year-old Rahman Haroun has been criticized by migrants' rights groups.
The Sudanese man was arrested by British police close to the Folkestone terminal in Kent.
Vessels led by anti-submarine ship on way to northern Atlantic for defense drills, says Russian navy.