Adding tags around “Read the rest of the story”

How is it possible to alter the anchor tags whenever a ‘cut-off’ is used.

For example, whenever I use ‘<!–more–‘ in my post I get the following anchor rendered:

<a href="https://website.com/?p=1#more-1">More</a>

What I would like to do is be able to add selector information, and even possibly use an image instead:

<a href="[post]"><img src="more.gif"></a>

Anyone have any tips for this?

Thanks,
Marc