Hikes around Alice Springs and Larapinta trail

Hi, I am hoping to go to Alice Springs to follow the Larapinta trail in June. I would like to explore as much of the area as I can. I have lots of time but I wont have many options for transport. Does anyone know of any other worthwhile trails/peaks in this area either out of Alice Springs or coming off the Larapinta? I am most interested in those areas accessible by public transport / bike / walking ? A hire car is a possibility for shorter trips. I would also be interested if anyone has any info on public transport out of Alice Springs to other good walking areas.