Is it only when you have it for breakfast?
If so reckon its mor4e likely to be either dawn phenomenon, foot on the floor or the sugar rather than the actual coffee
I know I get dawn phenomenon so my sugar level goes up between 3am and 7am no matter what, some get foot-on-the-floor and theirs go up when they get up from sleep, or it may just be you're very sensitive to sugar first thing
I know I do need more insulin in the morning for carbs than later in the day as well, and anything even vaguely sweet makes my sugar level rocket first thing- it does come back down to what it should be but takes its fine old time.