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Taxis don't seem to be the best way to get around Avignon at night. The best bet is to make sure you have a number of a taxi company and call them before you want to head home. That way you can wait in a bar or a coffee shop while they come to collect you.
Buses are frequent enough, but this is a relatively small town of c. 80,000 people so the bus network isn't necessarily as comprehensive or as you might like being a tourist. There is only one bus route for instance over to Villeneuve lez Avignon which doesn't run too frequently.