Whether you should upgrade really gets down to the actual model of your partner's Mac and what features you may want. Unfortunately your question doesn't elaborate on this.
However, on a more general note, a Mac running Snow Leopard 10.6.8 will have the Mac App Store app. This provides access to Apple and third party software. It's also how you would normally upgrade macOS.
Without knowing the specifics of your situation, you can run the App Store app to see whether you get the option to upgrade to Lion (i.e. OS X 10.7). Once you have upgraded to this, you can then upgrade to El Capitan.
However, what options you get will depend on the Apple ID being used and the actual model. You can sign into the Mac App Store by:
- Launch the App Store app on your Mac (it's inside the Applications folder).
- Go to the Store menu
- Select Sign In
- Enter the Apple ID and Password
Once you're logged in you can access any downloads made with that Apple ID. For example, if that Apple ID had been used to download El Capitan in the past (regardless of the computer), then you will see it listed in the Purchases tab and can download it again.
If it hasn't been used to download El Capitan, then you won't get the option to download it as it's not the current version of macOS. However, the App Store app will allow you to download the most recent version of macOS that will run on your Mac, so you may still be able to install El Capitan (you haven't provided details of the Mac model, hence the uncertainty here).
In summary, after updating to Lion you should get the option to update from that to another version of macOS, but the version you can update to will depend on your computer and potentially also on the purchase history of the Apple ID being used.
On a final note, whether you should update from Snow Leopard or not is a totally different question. Presumably your partner has been happy with Snow Leopard and may not want/need any of the 'new' features. Or they may have software installed that works well in Snow Leopard and not in later versions of the OS.
Either way, there is no cost involved in purchasing/upgrading macOS.