I don't see any reason to do an interim upgrade to ZC 1.5.4 especially since ZC 1.5.5f is already pretty much built.
Then once the PHP version has been increased to 5.6, ZC 1.5.6 can still operate on PHP 5.6 without issue so it too could be built/rebuilt in this environment without issue. Operating ZC 1.5.6 at php levels below 5.6 would require some "minor" code changes (though in many locations)
such as reverting array style notation (square brackets `[]`) back to `array()`. If I remember correctly that was pretty much the only change that was made for ZC 1.5.6 that basically locked it into the lowest version of 5.6 being supported. It's an important step in the right direction making it more possible to reap the coding benefits of php 7.x.
But, it seems that at least in the relative short term that you can get to ZC 1.5.6 on the current server. Then either the host will move to some form of php 7.x (to be at least somewhat current) or the account would need to move to a php 7.x hosting location. In either way, continued operation can occur on the current server and the upgrade process worked along the way as necessary.
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