One of the most common anticipated difficulties in applying mainstream maximum likelihood inference upon extreme values is articulated on the scarcity of extreme observations for bringing the extreme value theorem to hold across a series of maxima. This paper introduces a new variant of the Lq-likelihood method through its linkage with a particular deformed logarithm which preserves the self-dual property of the standard logarithm. Since the focus is on relatively small samples consisting of those maximum values within each sub-sampled block (by splitting the sample into blocks of equal length), the maximum Lq estimation will favour reducing uncertainty associated with the variance leaving the bias unchallenged. A comprehensive simulation study demonstrates that the introduction of a more sophisticated treatment of maximum likelihood improves the estimation of extreme characteristics, with significant implications for return-level estimation which is a crucial component in risk assessment for many operational settings prone to extreme hazards, such as earthquakes, floods or epidemics. We provide an illustrative example of how the proposed tilting of Lq-likelihood can improve inference on extreme events by drawing on public health data.