If you move stealthily, you do not want to be seen or felt. A cat moves stealthily when it approaches a mouse.
Stealthily is the adverb form of stealth: Think stealth bomber, that miraculous, if flawed, creation of the American military that was supposed to escape detection, including being nearly invisible on radar. We often think of hunters as moving stealthily. They sneak up on their prey, soundlessly. Originally, the word implied that you were looking to rob or steal something, which makes sense: this term is all about being secretive and undetected. Clumsy people have trouble doing things stealthily.
in a stealthy manner“stealthily they advanced upstream”