It is well known that the existence of outliers in a sample can significantly affect the estimation of population parameters. Intuition suggests that this is even more the case in the context of small area estimation. If influential, outliers may heavily affect parameter estimates for areas in which they occur, especially when the domain-sample size is tiny. An obvious remedy is to use robust estimators but with the drawback of a potential bias. We compare different approaches, including some new ones, for bias calibration in this context. Among other findings, the simulations indicate that the new proposals can lead to more efficient estimators compared to existing methods. We conclude the study applying our estimators to obtain Gini coefficients in labour market areas of the Tuscany region of Italy. The new methods reveal a different picture than existing methods. We extend our ideas to predictions for non-sampled areas.