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Variance-based sensitivity indices have established themselves as a reference among practitioners of sensitivity analysis of model outputs. A variance-based sensitivity analysis typically produces the first-order sensitivity indices $S_j$ and the so-called total-effect sensitivity indices $T_j$ for the uncertain factors of the mathematical model under analysis. The cost of the analysis depends upon the number of model evaluations needed to obtain stable and accurate values of the estimates. While efficient estimation procedures are available for $S_j$, this availability is less the case for $T_j$. When estimating these indices, one can either use a sample-based approach whose computational cost depends on the number of factors or use approaches based on meta modelling/emulators. The present work focuses on sample-based estimation procedures for $T_j$ and tests different avenues to achieve an algorithmic improvement over the existing best practices. To improve the exploration of the space of the input factors (design) and the formula to compute the indices (estimator), we propose strategies based on the concepts of economy and explorativity. We then discuss how several existing estimators perform along these characteristics. We conclude that: a) sample-based approaches based on the use of multiple matrices to enhance the economy are outperformed by designs using fewer matrices but with better explorativity; b) among the latter, asymmetric designs perform the best and outperform symmetric designs having corrective terms for spurious correlations; c) improving on the existing best practices is fraught with difficulties; and d) ameliorating the results comes at the cost of introducing extra design parameters.