Application of ribozymes for knockdown of RNA targets requires the identification of suitable target sites according to the consensus sequence. For the hairpin ribozyme, this was originally defined as Y⁻² N⁻¹ *G+¹ U+² Y+³ B+⁴, with Y = U or C, and B = U, C or G, and C being the preferred nucleobase at positions -2 and +4. In the context of development of ribozymes for destruction of an oncogenic mRNA, we have designed ribozyme variants that efficiently process RNA substrates at U⁻² G⁻¹ *G+¹ U+² A+³ A+⁴ sites. Substrates with G⁻¹ *G+¹ U+² A+³ sites were previously shown to be processed by the wild-type hairpin ribozyme. However, our study demonstrates that, in the specific sequence context of the substrate studied herein, compensatory base changes in the ribozyme improve activity for cleavage (eight-fold) and ligation (100-fold). In particular, we show that A+³ and A+⁴ are well tolerated if compensatory mutations are made at positions 6 and 7 of the ribozyme strand. Adenine at position +4 is neutralized by G⁶ →U, owing to restoration of a Watson-Crick base pair in helix 1. In this ribozyme-substrate complex, adenine at position +3 is also tolerated, with a slightly decreased cleavage rate. Additional substitution of A⁷ with uracil doubled the cleavage rate and restored ligation, which was lost in variants with A⁷, C⁷ and G⁷. The ability to cleave, in conjunction with the inability to ligate RNA, makes these ribozyme variants particularly suitable candidates for RNA destruction.