Once Deepfathom Echo becomes a copy of another creature, it no longer has its triggered ability until end of turn.
Deepfathom Echo
Once an ability that causes a creature to explore begins to resolve, no player may take any other actions until it's done. Notably, opponents can't try to remove the exploring creature after you reveal a nonland card but before it receives a counter.
In some unusual cases, noncreature permanents may explore. For example, if the creature card returned by Defossilize is somehow not a creature once it's on the battlefield, it can still explore. You'll take all the same actions, and you may end up putting a +1/+1 counter on the permanent. (Note that some effects target a creature, and those effects would still require a legal target to have it explore.)
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Matt Stewart
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
