Good point. That is one important thing to consider if you decide upon a EDC lifestyle, especially EDG of a gun. First - it is a lifestyle. It is not just something you do once in a while, when you feel like it, it is a commitment. Secondly - especially with a gun, is it is most likely NOT going to be comfortable. That's right, it is probably going to be an inconvenience - to you, with your clothes, with your holsters, etc. EDC will always force you to decide what you're carrying and what clothes you will have to wear to help conceal it. In your case, it will also help you determine if you want to carry an EDC bag. Perhaps you will only want to carry your gun in your purse*, and the rest of your EDC in a dedicated EDC bag. Sure, an extra bag is a pain, but again, this is a choice you are making to be prepared.
Part of the fun then becomes picking out your EDC. If you're trying to keep it small, there are lots of options today for EDC. For example, you might consider products that
Swiss+Tech Products - Multi-Purpose Key Ring Tools carry. They have all sizes of multi-tools from micro tools, to small multi-tools, to large multi-tools. Once you think about how you may use a certain tool, then you can go about looking for a small version of a tool that will get the job done.
Some of their various sizes of tools:
Microtools
Swiss+Tech Products - Multi-Purpose Key Ring Tools
Small multi-tools
Swiss+Tech Products - Multi-Purpose Key Ring Tools
Large Multi-tools
Swiss+Tech Products - Multi-Purpose Key Ring Tools
And that is just part of their catalog.
* Carrying your weapon off-body has its own limitations and dangers. Be sure to fully research off-body carry.