To get around this without a converter, simply don't plug the yellow composite video cable from your console into the yellow composite video jack on your switchboxes. Instead, plug it into the red, green, or blue component input of the switchbox that is being used for the composite input on your tv. Daisy chain switchboxes as necessary to get enough inputs. They don't care about what color the jack on the back is. Another option is to dedicate a second switchbox to composite console's and route the yellow composite output from it to the corresponding component connection on your first switchbox.
For another example of a way to be clever to get more out of a switchbox, I have a four port switchbox with four S-Video connected consoles plugged in with the appropriate outputs connected to the corresponding connections on the rear of my Trinitron. And then with the three rear inputs on the rear that also include component connections, I have three composite connected console's plugged in using the red/green/blue inputs with the outputs routed to a composite connection on the rear of the tv. Works just fine and you just have to remember which color corresponds to correct color as you're plugging everything in. So instead of four console's on this four way switchbox, I instead have 7 and eliminated having to connect another switchbox.
Even my VHS/DVD recorder that supports composite, S-Video, and component (But strangely, just coaxial out and no coaxial in) isn't able to take a source from composite or S-Video and output it via component. Has to be entering as component or has to be a DVD being played back. So a route like that isn't a option to easily convert different sources to component sadly.
That was my first thought for you until I went and gave it a try just now. I used to do something similar 15 years ago with a VHS player when I didn't have a tv with a composite input to get around having to buy RF switches for console's like the Nintendo 64 that came with composite cables out of the box. But no go here so I doubt you could achieve this with something you possibly already even own.
And I suspect you won't get terribly pleasing results with a cheap device designed to convert a composite signal to component, but I don't know for sure. I'm not even sure I've ever seen such a device although I'm sure they exist. Maybe one of the tech gurus here like Darklordjames can't chime in with some information about such things.
Of course a quality external scaler would do it for you. But besides the expense, they come with their own fair share of issues. Not something that I think is quite there yet for the classic gamer in terms having a good balance of quality, flexibility, and affordability. They're still very much niche devices and can do an excellent job at times in some circumstances but fail miserably with a different console.