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A popular destination, as an alternative to the usual Gibraltar sights, is a day trip to Tanger in Morocco.
Tanger is easier to reach than one may imagine; there is a ferry port in Gibraltar as one option but the crossings only operates on Friday at 22:00 and returns back to Gibraltar from Tanger on Sunday at 19:00. The service is run by Turners of Irish Town and this is their only service during the week. The following website has information on timetables and fares: http://www.frs.es/ If the weather is bad, the service will be cancelled without notification.
You may wish to consider the option of doing a day trip with one of the Gibraltan tour operators such as Blands who have an office on Main Street. They will arrange pick up from your hotel and transportation through the Spanish border to the other option which may be Tarifa in Spain. The only drawback here is that such excursions are only offered once a week with Blands although others may offer a daily service. It helps to shop around when in Gibraltar.
A seasoned traveller doing the Tanger crossing may make their way to Algeciras ferry port simply by crossing the border into La Linea in Spain and walking (or getting a taxi if you have luggage) to the bus station. From here a bus can be got for a few €'s to Algeciras which has hourly crossings to Tanger but takes over two hours. The alternative, as said, is Tarifa which is geographically closer to Tanger, takes an hour to cross; ferry crossings are at two hour intervals. As a matter of interest, Tarifa has old Roman ruins which can be explored to pass the time waiting for the ferry.