You can do what I do (My swap magic 3.6 dvd never worked to begin with, laser issue I suspect).
Burn a copy of CDLoader.
1) Load Swap Magic CD
2) Swap in CDLoader with Slide tool or whatever, and Load it
3) Once CDLoader is running, Hit Eject
4) Put a retail DVD game in, perferably something with a large TOC
5) Let the game load, it should return back to the screen where it told you to push X. In the lower left part of the screen it should say "PS2 DVD" this lets you know you are now on a DVD TOC.
6) On my version of CDLoader I had to press X now.
7) Screen will go red and say "Insert DVDR with EA method and Press X"
8) Swap in your backup and press X.
It should load but if it goes back to browser at this point then you might have laser issue (or bad media, I don't know what you are using), you might try laser cleaning or laser adjustment.
I don't know the Slim's methods very well but just use whatever method you have to swap in the disc at the times mentioned. Ie: Magic Keys, Switch, etc. You just have to get the back up and CDLoader in without the system realizing you made the switch.
Quote: Q.)Swap Magic DVD Disc Wont Boot my DVD Backup.
A.)1.Swap Magic DVD does not work so well for some games,instead of useing the Swap Magic DVD disc just use the CD instead, It Works bettter.
To use the Swap Magic CD to boot DVD Games Do the Following:
1.Eject the CD tray and insert the Swap Magic CD Disc.
2.Reset your PS2 and wait till Swap Magic loads Up
3.When you see the Swap Magic Screen that saids "Load Game" along with the video settings(PAL or NTSC) eject the PS2 cd tray and take out the Swap Magic CD and and the screen will now say Refresh TOC
4.Now insert a PS2 DVD Original Game(Silver Bottom) With a High TOC(check your ps2 games on the computer and see which is the biggest game you have in size)
4.after it goes back to the Swap Magic Screen Open the PS2 Flip Top Cover and insert the DVD Backup, After This The screen should turn red and the game will start.
Which I guess in your case, following the directions starting from the point where you get Swap Magic to run.