Ranked set sampling is a well-known and efficient method compared to simple random sampling for estimating population parameters. In this study, we focus on the challenge of estimating the scale parameter of the primary variable $Z$ using a multistage ranked set sample obtained by ordering the marginal observations of an auxiliary variable $W$, where the pair $(W,Z)$ follows the Farlie–Gumbel–Morgenstern bivariate Bilal distribution. Assuming that the dependence parameter $\varphi$ is known, we introduce the best linear unbiased estimator for the scale parameter of the primary variable, utilizing a multistage ranked set sample. We also compare the efficiency of the proposed estimator with that of the maximum likelihood estimator based on the same number of measured units. It is found that the suggested estimators are more efficient than the classical estimators considered in this study.