You did not provide enough information. But, you gave a few vague clues. I will attempt to guess the configuration. You describe a coupling between the compressors. I assume you have a single driver with two compressor cases. A centrifugal compressor will normally have the thrust bearing at the non-drive end. The compressor case will be pinned at the drive end and have guide keys that allow it to grow away from the drive coupling. The case grows away from the coupling. The shaft grows toward the coupling. The two somewhat cancel each other out. This works great for a single driver driving a single compressor.
If you add a second compressor case driven off the back end of the first one, you have a challenge. The thermal growth of the first compressor is all directed at that second coupling. Depending on the length of the first compressor and the temperature it runs, this can easily be several millimeters. To accommodate this movement you either have to pre-stretch the coupling, or use a gear or spline coupling that allows for that much movement.
Provide the configuration of your machine train with the locations of the thrust bearings and the case keys and pins along with the process temperatures if you want a more detailed explanation.
Johnny Pellin