Hi Jessie
I wouldn't take anymore background insulin, you do need to keep a eye on your BG's to ensure that they do increase to far.. The problem you would face to take extra background insulin now to compersate is unlikely to make much difference, if for any reason you need to make a adjustment on background insulin then it can take a couple of days to show up how this has really effected everything...
The purpose of background insulin is to cover the glucose that is going into the blood from your liver only...
I am surprised what you have been told by your DSN as background insulin isn't designed to cover any food at all, if you can eat snacks between meals and the background is covering then this can indicate that your injection of background insulin is wrong...
Background insulin should in theory hold your BG's steady with a 2mmol/l +/- for fasting periods of normal activity... If the snack you were eating is 10g of carbs or less yes most diabetics wouldn't actuall inject quick to compensate..
I would ask your DSN if there is a carb counting course available, as this will teach you how to adjust your insulin to your carbs, how to ensure that you are taking the correct background dose etc, how and when to take action to ensure good control of your diabetes.