You did see him with a black cap. It was the same cap really : Navy regulations said that during « summer » (beginning… May 1st if I’m not mistaken!), the black sailors’ caps (and Navy officers' caps too) had a white cap cover which covered only the top part of the cap.
Here is the original cap without it’s cover. This cap came from the Alexander Palace so it may have belonged to the tsarevich.

Clockworkgirl21 : until World War I broke out, Alexis did wear his sailor uniform from the « Standart » very often, even on land. It’s the uniform he wore the most often, probably almost everyday. Here is a photo showing him on horseback wearing the sailor‘s cap with the white summer cover.

Sadly enough, he wasn’t allowed to ride on horseback but he did have his picture taken as if he was!