Antiviral restriction factors such as MxA (myxovirus resistance protein A) inhibit a broad range of viruses. However, they face the challenge of maintaining this breadth as viruses evolve to escape their defense. Viral escape drives restriction factors to evolve rapidly, selecting for amino acid changes at their virus-binding interfaces to regain defense. How do restriction factors balance the breadth of antiviral functions against the need to evolve specificity against individual escaping viruses? We explored this question in human MxA, which uses its rapidly evolving loop L4 as the specificity determinant for orthomyxoviruses such as THOV and IAV. Previous combinatorial mutagenesis of rapidly evolving residues in human MxA loop L4 revealed variants with a ten-fold increase in potency against THOV. However, this strategy did not yield improved IAV restriction, suggesting a strong tradeoff between antiviral specificity and breadth. Here, using a modified combinatorial mutagenesis strategy, we find 'super-restrictor' MxA variants with over ten-fold enhanced restriction of the avian IAV strain H5N1 but reduced THOV restriction. Analysis of super-restrictor MxA variants reveals that the identity of residue 561 explains most of MxA's breadth-specificity tradeoff in H5N1 versus THOV restriction. However, rare 'generalist' super-restrictors with enhanced restriction of both viruses allow MxA to overcome the breadth-specificity tradeoff. Finally, we show that a heterozygous combination of two 'specialist' super-restrictors, one against THOV and the other against IAV, enhances restriction against both viruses. Thus, two strategies enable restriction factors such as MxA to increase their restriction of diverse viruses to overcome breadth-specificity tradeoffs that may be pervasive in host-virus conflicts.