Previous approaches to the problem of transient scattering by conducting bodies have utilized the well-known marching-on-in-time solution procedures. However, these procedures are very dependent on discretization techniques and in many cases lead to instabilities as time progresses. Moreover, the accuracy of the solution procedure cannot be verified easily and usually there is no error estimation. Recently an alternate approach to the solution of transient scattering by thin wires was presented based on the conjugate gradient (CG) method. In this procedure, space and time are discretized independently into subintervals and the error is minimized iteratively. Unfortunately, this procedure is very slow, not easily extendable to other geometries, and moreover, some of the advantages of marching-on-in-time are lost. In this paper, again the conjugate gradient method is applied to solve the above problem, but this time, reducing the error to a desired value at each time step. Since the error is reduced at each time step, marching-on-in-time can still be done without error accumulation as time progresses. Computationally, this procedure is as fast as conventional marching-on-in-time. Thus, this new method retains all the advantages of marching-on-in-time and yet does not introduce instabilities in the late time. It is also possible to apply this procedure to other geometries. Details of the solution procedure along with numerical results are also presented.