It's not really an exception so much an application of the "reasonable" part, right? You don't have a right against search and seizure, just unreasonable search

It's not really an exception so much an application of the "reasonable" part, right? You don't have a right against search and seizure, just unreasonable searches and seizures. The existence of a warrant makes a search definitely reasonable, but there are contexts where a search can be reasonable without one.