It is believed two men got into the back of the lorry at around 6.20am on Thursday, July 16, while it was in a layby.
The truck was driven to the Tesco Express in Harrier Close, Calne. It is thought others arrived to help unload the cigarettes into another vehicle while the lorry was parked in a loading bay outside the store.
A Wiltshire Police spokesman said: “We are appealing for anyone who may have been in the area at the time to try to recall if they noticed anything suspicious. We are particularly keen for any dashcam footage recorded by anyone who may have been near the layby at Derry Hill or the Tesco store in Calne at the time."
Information can be reported by calling 101 or anonymously, via Crimestoppers, on 0800 555 111.