Speech enhancement (SE) is used as a frontend in speech applications including automatic speech recognition (ASR) and telecommunication. A difficulty in using the SE frontend is that the appropriate noise reduction level differs depending on applications and/or noise characteristics. In this study, we propose "signal-to-noise ratio improvement (SNRi) target training"; the SE frontend is trained to output a signal whose SNRi is controlled by an auxiliary scalar input. In joint training with a backend, the target SNRi value is estimated by an auxiliary network. By training all networks to minimize the backend task loss, we can estimate the appropriate noise reduction level for each noisy input in a data-driven scheme. Our experiments showed that the SNRi target training enables control of the output SNRi. In addition, the proposed joint training relatively reduces word error rate by 4.0\% and 5.7\% compared to a Conformer-based standard ASR model and conventional SE-ASR joint training model, respectively. Furthermore, by analyzing the predicted target SNRi, we observed the jointly trained network automatically controls the target SNRi according to noise characteristics. Audio demos are available in our demo page: google.github.io/df-conformer/snri_target/.