Not necessarily. File transferring part is the matter of few minutes. The whole moving process usually takes a couple of days to complete (the longest part being propagation which can take between 24 to 72 hours).
During the transition, whereas some end users may be visiting your website at its new location, others may still be receiving old information and are being routed to your previous host (as long as your files are still active on their servers). Therefore, it is best to continue service with your old hosting provider until your domain name has been completely propagated and is pointing to UnicHost’s servers.