You can always get yourself some good bike panniers like Ortlieb rear pannier or Ortlieb bike pannier or any other rear bike bag that you believe will carry your clothes safely to the other side of the city. Such accessories are essential for people who commute on a bike to work, the gym, or on weekend bike tours. Always keep a pair or two of clothes and if it is winter season you may want to pack some warmers along with everything else.