A new apparatus for coincidence velocity mapping has been developed for studies on the inner-shell photoionization dynamics of molecules. It enables velocity imaging of electrons and ions simultaneously, and maintains time focusing. The characteristic feature of the apparatus is its applicability to a wide range of photoelectron kinetic energy from a few to 100 eV. To demonstrate the performance of the new apparatus, photoelectron angular distributions from Ne atoms and randomly oriented NO molecules, photoion angular distributions from CO molecules, and photoelectron angular distributions from fixed-in-space CO and NO molecules, have been examined. In the data processing procedures, a new peeling method has been developed to construct a three-dimensional photoelectron angular distribution, which is not cylindrically symmetric, from a two-dimensional one.