There isn't a single perfect adverb for "explore". Here's why and some options depending on the context:

Exploring needs more context: "Explore" describes an action, but the specific way it's done needs clarification. Do you want to convey:

* Thoroughness? "Thoroughly explore"

* Freedom? "Freely explore"

* Curiosity? "Curiously explore"

* Method? "Methodically explore", "carefully explore", "boldly explore"

* Result? "Successfully explore", "fruitfully explore"

Alternatives: Sometimes, verbs that express the *way* of exploring work better:

* Investigate: "They investigated the area thoroughly."

* Examine: "She examined the options carefully."

* Search: "He searched for clues diligently."

Ultimately, the best adverb depends on the specific meaning you want to convey.