GANStrument, exploiting GANs with a pitch-invariant feature extractor and instance conditioning technique, has shown remarkable capabilities in synthesizing realistic instrument sounds. To further improve the reconstruction ability and pitch accuracy to enhance the editability of user-provided sound, we propose HyperGANStrument, which introduces a pitch-invariant hypernetwork to modulate the weights of a pre-trained GANStrument generator, given a one-shot sound as input. The hypernetwork modulation provides feedback for the generator in the reconstruction of the input sound. In addition, we take advantage of an adversarial fine-tuning scheme for the hypernetwork to improve the reconstruction fidelity and generation diversity of the generator. Experimental results show that the proposed model not only enhances the generation capability of GANStrument but also significantly improves the editability of synthesized sounds. Audio examples are available at the online demo page.