I am affraid you won't find a perfect solution here - to provide visibility and affordance (in a visual world) you need to keep some element visible. Showing more on hover is a good idea provided this is a desktop solution (you mentioned Chrome Web Store).
There are pros and cons of both solutions you have mentioned:
Triggering on click will make user click it to discover/rediscover the hidden features (users may avoid clicking it as they forget "what was the icon used for?" and not willing to trigger some unknown action. In other words - this icon will need them to think. It's no good except tools that are used on a very daily basis, where user remembers what is a tool for, or if an icon is very, very self-defineable and understandable.
Triggering on hover is different. Good about it is that user will sometimes trigger the extended options accidentally, which will remind him about what's in there. But at the same time, it needs proper approach because this can become too annoying, resulting finally in user uninstalling the extension. But it cannot interfere with what's on a website user is visiting, limiting user experience of the website.
Thus, you need to properly plan where the initial trigger will be placed, and optionally you can also let user customize this position. One of my favourite examples here is Jing (http://www.techsmith.com/jing.html), where a little on-hover spot is glued to a side of a screen, but its position can be customized (user can drag it around, glueing it to various places around the screen. Except Chrome, which uses the very top of the interface on Windows, the best place for it is top of the screen (it resides just above the main application window content, just over the title bar). But since I use Chrome, the good place is leftmost side of the screen now. It's because most of the pages are not full-width, and as such they are usually centered. Usually. For all the rest, the drag-me-somewhere-else feature does the job perfectly. So as your solution will be used in a context of visiting websites, I would make it the default position, informing user (during onboarding just after the extension installation) that s/he can drag it somewhere else if needed.
Secondly, I would focus on:
- a great icon, providing correct and fast recognition of what the trigger spot is for, and what treasures it hides. Hamburger menu is a very bad choice here, as if you take a browser as a whole, hamburger menu is contextual, specific for a visited site and allows accessing the main site navigation.
- making the hover spot big enough, so that user does not need to focus on placing his cursor over it very precisely - one clue is that you can make the hover area bigger than its visual indicator. But don't go too wild here, occupying the full left border of the window, as it may render some websites unusable below, even though you provide the drag around feature.
One more thing is that if your solution is content-contextual, you may go a little bit different way, showing contextual popup menu e.g. after text selection (you may check Diigo extension as a reference).