Inverting genetic logic gates fueled by transcriptional repression is an established building block in genetic circuit design. Often, the gates' dose-response curves require large changes in dose to transition between logic ON and OFF states, potentially leading to logically indeterminate intermediate states when gates are connected. Additionally, leakage in the OFF state is a general concern, especially at the output stages of a circuit. This study explores the potential to improve inverting logic gates through the introduction of an additional sequestration reaction between the input and output chemical species of the gate. As a mechanism of study, we employ antisense RNAs (asRNAs) expressed alongside the mRNA (mRNA) of the logic gate within single transcripts. These asRNAs target mRNAs of adjacent gates and create additional feedback that supports the protein-mediated repression of the gates. Numerical and symbolic analysis indicates that the sequestration steepens the gate's dose-response curve, reduces leakage, and can potentially be used to adjust the location of logic transition. To leverage these effects, we demonstrate how design parameters can be tuned to obtain desired dose-response curves and outline how arbitrary combinational circuits can be assembled using the improved gates. Finally, we also discuss an implementation using split transcripts.