You can absolutely create a new tab. I just created an example setup for something like this here
I use a new sheet (Sheet2) that contains extra fields as well as the original data imported through a =query statement in Task2!B1
The query is very simple: =query('Form Responses 1'!A1:Z, "select A,B") - add more than A,B if you are adding more columns (and if you're adding way more columns, you will probably want to replace Z with something larger.)
It is possible you could even insert the extra column directly in the Form Responses sheet; I've not tried what happens if you add columns there.
