Couldn't someone with a ship just go on their own trip without the mandate of a King and just go exploring? That's what people did back then didn't they? They'd explore?
Maybe some odd objects found in America with distinctly Roman Anglo styles and features which predate known trips to the Americas were from people who visited without telling anyone.
People who just crossed the Atlantic on their own.
???
In Salt: a World History, Mark Kurlansky suggests that cod fishermen could have discovered Newfoundland and its massive cod fisheries as early as the mid 15th century. The reasoning for not relaying this discovery would be the intense competition for fishing spots between rival companies. Whether or not the theory holds ground is up for debate but the idea is interesting and should be noted. For more earlier discoveries see The Vinland Sagas for the Norse discovery of America which is very well documented, with archeologists finding their temporary settlement in L'Anse aux Meadows in Newfoundland
There's always room for discussion, but perhaps the section Travel and contact across the Atlantic before Columbus in our FAQ will answer your inquiry.