Any proceedings for street work heretofore commenced under any method or system provided by any Charter of said City and County, or by act of the Legislature, or by ordinance of said City and County, shall not be affected by this Article but may be continued until completion by and under the method provided by the Charter, law or ordinance under which they were originally commenced, or they may be abandoned by order of said Director of Public Works and instituted under the proceedings provided for in this Article, in case no contract for the work or improvement has been entered into.