Temporal envelope is the primary acoustic cue used in most cochlear implant (CI) speech processors to elicit speech perception for patients fitted with CI devices. Envelope compression narrows down envelope dynamic range and accordingly degrades speech understanding abilities of CI users, especially under challenging listening conditions (e.g., in noise). A new adaptive envelope compression (AEC) strategy was proposed recently, which in contrast to the traditional static envelope compression, is effective at enhancing the modulation depth of envelope waveform by making best use of its dynamic range and thus improving the intelligibility of envelope-based speech. The present study further explored the effect of adaptation rate in envelope compression on the intelligibility of compressed-envelope based speech. Moreover, since noise reduction is another essential unit in modern CI systems, the compatibility of AEC and noise reduction was also investigated. In this study, listening experiments were carried out by presenting vocoded sentences to normal hearing listeners for recognition. Experimental results demonstrated that the adaptation rate in envelope compression had a notable effect on the speech intelligibility performance of the AEC strategy. By specifying a suitable adaptation rate, speech intelligibility could be enhanced significantly in noise compared to when using static envelope compression. Moreover, results confirmed that the AEC strategy was suitable for combining with noise reduction to improve the intelligibility of envelope-based speech in noise.