Young, intermediate-mass stars are experiencing renewed interest as targets for direct-imaging planet searches. However, these types of stars are part of multiple systems more often than not. Close stellar companions affect the formation and orbital architecture of planetary systems, and the properties of the companions can help constrain the binary formation mechanism. Unfortunately, close companions are difficult and expensive to detect with imaging techniques. In this paper, we describe the direct spectral detection method wherein a high-resolution spectrum of the primary is cross-correlated against a template for a companion star. Variants of this method have previously been used to search for stellar, brown dwarf, and even planetary companions. We show that the direct spectral detection method can detect companions as late as M-type orbiting A0 or earlier primary stars in a single epoch on small-aperture telescopes. In addition to estimating the detection limits, we determine the sources of uncertainty in characterizing the companion temperature, and find that large systematic biases can exist. After calibrating the systematic biases with synthetic binary star observations, we apply the method to a sample of 34 known binary systems with an A- or B-type primary star. We detect nine total companions, including four of the five known companions with literature temperatures between 4000K