For emphasis:
* Strongly: "He strongly suggested we leave early."
* Warmly: "She warmly suggested we try the new restaurant."
* Warmly: "The teacher gently suggested I try a different approach."
* Politely: "The waiter politely suggested a different wine."
For indicating the manner of suggestion:
* Carefully: "She carefully suggested a solution."
* Directly: "He directly suggested we meet tomorrow."
* Casually: "He casually suggested we go for a walk."
* Tentatively: "She tentatively suggested a new plan."
For specifying the effect:
* Persuasively: "He persuasively suggested a new strategy."
* Effectively: "She effectively suggested a change of plans."
It's also possible to use adverbs that describe the speaker or their intention:
* Hesitantly: "He hesitantly suggested going to the party."
* Confidently: "She confidently suggested a new approach."
* Enthusiastically: "He enthusiastically suggested trying the new recipe."
Ultimately, the best adverb to use will depend on the context and the meaning you want to convey.