The "drift-and-minorization" method, introduced and popularized in (Rosenthal, 1995; Meyn and Tweedie, 1994; Meyn and Tweedie, 2012), remains the most popular approach for bounding the convergence rates of Markov chains used in statistical computation. This approach requires estimates of two quantities: the rate at which a single copy of the Markov chain "drifts" towards a fixed "small set", and a "minorization condition" which gives the worst-case time for two Markov chains started within the small set to couple with moderately large probability. In this paper, we build on (Oliveira, 2012; Peres and Sousi, 2015) and our work (Anderson, Duanmu, Smith, 2019a; Anderson, Duanmu, Smith, 2019b) to replace the "minorization condition" with an alternative "hitting condition" that is stated in terms of only one Markov chain, and illustrate how this can be used to obtain similar bounds that can be easier to use.