Mrs Jeanes, 44, said: "We moved here 19 months ago and our house backs onto the bus station.
"He is an old boy and is very friendly. At first Dodger kept going to the bus station because people there fed him tit-bits and scraps of food.
"But then he started climbing on board the buses because they are almost like greenhouses when it is sunny.
"Then last week I found out he had travelled to Charmouth and back, which is a 10 mile round trip.
"I hadn't seen him all morning until my daughter Emily told me one of her friends had just seen him on the bus at Charmouth.
"I couldn't believe it and panicked. I got into my car to go off and look for him and then at that moment the bus pulled up near our house and low and behold he got off.
"He had fallen asleep on board and nobody knew about it. When the driver realised he knew who Dodger was and where he lived and kept him on board.
"That afternoon I saw Dodger climb on board another bus and I rushed to tell the driver.
"I was shocked when she told me Dodger was always on there and liked to sit on the seats because they are warm from where people have been sitting.
"The drivers buy cat food for him and he sits on people's laps.
"Sometimes he just sits in the middle of the road and waits for the bus to turn up before he gets on."
Dodger is familiar to regular bus passengers and drivers, but Mrs Jeanes still receives several calls a week asking if she has lost a ginger cat.
A spokesman for bus firm First said they didn't mind Dodger on their buses but didn't actively encourage him.
He said: "The drivers have been asked not to feed it because we recognise that cat has an owner and we do not want to discourage it from returning home for food and shelter, but in principle we do not have a problem with it being around the bus station.
"Given this cat is elderly we suspect it would be eligible for free travel, perhaps a bus puss, if such a thing existed."