Experimental design and analysis is an effective and commonly used tool in scientific investigations and industrial applications. Orthogonal arrays, such as factorial and fractional factorial designs, are popular experimental plans for identifying important factors. Motivated by an antiviral drug experiment, we introduce a new class of composite designs based on a two-level factorial design and a three-level orthogonal array. These designs have many desirable features and are effective for factor screening and response surface modeling. Some advantages are that they can use resolution IV designs in the screening stage, they can perform in-depth analyses, and they can be used in either a single or a sequential experiment. We study the construction method and compare the new composite designs with existing ones. We illustrate the methodology with data from an experiment that studies the effects of five antiviral drugs on the Herpes simplex virus type 1.