A solid-color scarf leaves more room for your face than a large pattern. Pick a calm tone that matches your blouse or blazer. Choose a background that differs from the fabric so the outline of your head stays visible. A dark scarf against a dark wall can merge with the background in a small image.
Lay your outfit and scarf side by side before taking the photo. Check the combination in daylight, since fabric can look different under a room lamp. Matte material avoids strong reflections. For more clothing guidance, see what to wear for a headshot.
Tie the scarf the way you would wear it at work. Set the edge at your forehead and arrange the fabric under your chin before taking the selfie. Your face should be clearly visible. If you upload a selfie wearing your hijab, this is the wrap shape you should compare with the result.
Stand at a window with light coming from the front or slightly from the side. A ceiling lamp alone often casts shadows under your eyes and chin. Hold the camera at eye level with some space above your head. Leave room around your head for LinkedIn's circular crop, and check the crop in the finished PDF when using the photo in a CV.