If you are proven innocent in court can you confess to the crime?

In countries observing the rule of double jeopardy, a person cannot be tried twice for the same crime.
Even if you confess to the crime after the trial you cannot be tried again for this, however, there are countries where the double jeopardy law has been removed.
I'm in the U.K. and 4 years ago they changed this law and started trying people for crimes they got away with years ago (where later evidence showed them to be guilty) of course this also applies to recent cases, some people have walked free from court only to be arrested on the steps of the court and charged, tried (some time later) and imprisoned for the same crime.