Does it look like you are loosing styling in https? Like maybe a css sheet isn't loading?
Have you tested your design, both http, https in other browsers.
My guess is, when it comes to links showing up in https, that they are actually there in http, but because of css being loaded correctly in http, they are either in a different location, or being hidden by another element, so you usually didn't see them, even though they are there. You can verify this in the source code.