Sequential sampling techniques are very useful when there is a significant cost in obtaining observations. Also, certain kinds of statistical inference can only be carried out under sequential procedures. Such examples include fixed-width confidence intervals for unknown population parameters. ^ Under a general setting, we modify the Stein-type two-stage methodology in order to incorporate some partial information in the form of a known and positive lower bound for an unknown nuisance parameter. Under certain conditions, this revised procedure is then shown to enjoy several second-order properties and expansions for the expectation of functions of the associated stopping variable. The theory, along with numerous sets of simulations, will show that our revised procedure is more precise than the original two-stage procedures which did not take into account any partial information. ^ The list of applications of our revised methodology is long. In particular, we discuss the construction of fixed-width confidence intervals and fixed-size confidence regions. Also, some point estimation problems where the risk function is minimized or bounded from above, and problems involving selection and ranking, are discussed. We demonstrate the broad range of problems which can be solved. We also show that it is easy to apply the revised procedure in practice.