Small transport First Officer reported that the aircraft entered Class D airspace prior to being handed off and ZME Center may not have been in contact with CGI Tower during the handoff. The reporter also noted that CGI Airport's operation hours may be inaccurate on the Chart Supplement.

Narrative

The flight pertaining to this report was scheduled to fly from ZZZ to CGI on Aircraft X at XA:00 local time. The flight plan was conducted at nine-thousand feet with three passengers to be dropped off in CGI. There were no discrepancies prior to departure. We departed ZZZ at XA:25 local time to CGI. There was nothing abnormal during takeoff or in cruise part of the flight. We were handed off to Memphis Center after Departure Control. Approximately thirty miles out; Center was letting us know where the airport was and asking us if we could see it. I was focusing on the location of the airport and was looking far out in the distance. Memphis Center instructed us to descend to 2500 feet prior to reaching the airport. We were still searching for the airport at this point and knew we were getting close. As we were both looking; we were growing closer to the airport and did not have its location until we were about to cross over midfield of the airport. Once we had the airport insight; we told Memphis Center. We were told that frequency change was approved; and we were cleared for the visual for Runway 28. However; we had entered the Class D prior to changing frequencies and handed off when we were already inside the Class D airspace. As we talked to Tower; we were instructed to fly eastbound and make a 180 to land on the visual for Runway 28. There were no traffic conflicts on this flight.It appeared that Memphis Center thought that Cape Girardeau was closed because we were instructed that we were cleared for the visual Runway 28 and never told us to contact Tower. We were only told frequency change approved. I think they thought they were handing us off on a CTAF frequency and not a Tower frequency. I looked in the Chart Supplement for the operation for Cape Girardeau Tower's hours. The Chart Supplement had the Tower operating at XB:00 local time. However; I searched online at airnav.com for the hours of operation; and it was reported that the Tower was open at XA:00 local time. This event occurred around XA:40 local time. If this observation is true; Memphis Center was not in contact with Cape Girardeau Tower prior to us entering their airspace. The action I can take for next time is: Have a situational awareness of how far I am from the airport and be ahead of the flight plan. Do not assume a handoff between Center and Tower or even Approach and Tower. Have the Tower or CTAF frequency on the standby communications. If the airport is not in sight within a certain distance; ask for an instrument approach to help be lined up with the landing runway.