A wide variety of estimators of the between-study variance are available in random-effects meta-analysis. Many, but not all, of these estimators are based on the method of moments. The DerSimonian-Laird estimator is widely used in applications, but the Paule-Mandel estimator is an alternative that is now recommended. Recently, DerSimonian and Kacker have developed two-step moment-based estimators of the between-study variance. We extend these two-step estimators so that multiple (more than two) steps are used. We establish the surprising result that the multistep estimator tends towards the Paule-Mandel estimator as the number of steps becomes large. Hence, the iterative scheme underlying our new multistep estimator provides a hitherto unknown relationship between two-step estimators and Paule-Mandel estimator. Our analysis suggests that two-step estimators are not necessarily distinct estimators in their own right; instead, they are quantities that are closely related to the usual iterative scheme that is used to calculate the Paule-Mandel estimate. The relationship that we establish between the multistep and Paule-Mandel estimator is another justification for the use of the latter estimator. Two-step and multistep estimators are perhaps best conceptualized as approximate Paule-Mandel estimators.