The cat has a Facebook page with over 9,500 members and also features in postcards promoting Dorchester.
After moving to the town in 2023 she settled in quickly and is known to visit more than 30 different shops across the town.
Ms Graves said: "It is a real balance between the joy that Susie brings with her adventures and keeping her safe.
"The admin team for her Facebook page work tirelessly and for free to help maintain that balance. Susie is, and I suspect always will be, a wanderer."
People are being asked to put Susie, who also lives with her brother Mr Waffles, back outside by 21:00 so she can go home to her family and have her dinner.
Ms Graves said this was "to ensure she knows her real home, so that if she is scared she knows where to come where she is safe".
"It also helps me keep an eye on her health and weight," she said.
"It would be great if people would remember she is a family pet."
Her recent antics include napping on a seat at a first aid course in a hotel after wandering in through an open door, visiting a department store and the town's barbers and cinema.
Susie's collar is also fitted with a tracker so her owner can keep an eye on her whereabouts.
