Safari - How to search on a web page from the current selection onwards
You can easily search from the point of selection using the following approach.

Immediately after selecting the string to search, use the keyboard shortcut Command + E to record the search string into find buffer.

Now press the keyboard shortcut Command + G to search from the selection point in the forward direction. To search backward, use the keyboard shortcut Command + Shift + G.

This works in most other apps apart from Safari.