The phrase "extra suitcase" refers to having additional things or responsibilities to handle, beyond what is normal or expected. It suggests having more than you can easily manage or carry. Full definition
Now I was returning her suitcase but thought it would be a good idea to have my own extra suitcase with me. (nofearoffashion.com)
Had to pay for the extra suitcase as it was... so now I too am dreaming of Masa Harina, deciding whether to pay 8 CHF for a tiny bag of it here.. (davidlebovitz.com)
I'll need an extra suitcase just for that, I'm sure haha! (101cookbooks.com)