The paper extends recent work on a modified Smith predictor strategy, which leads to significant improvements in its regulatory capacities for reference inputs and disturbances. High-order or long dead time stable, integrating and unstable plants are modelled as lower-order plant models with a longer time delay. The controllers are designed so that the delay-free component of the output is tuned to be either a first- or second-order response if there are no modelling errors in the assumed plant transfer function. Plant model transfer functions and the controller parameters are estimated using exact analysis from the peak amplitude and frequency of the process output obtained from a single-relay feedback test. Illustrative examples show the simplicity and superiority of the proposed controller design method over previously published approaches both for the setpoint response and for the load disturbance rejection.