Sugars rising in the morning pre carbs is also known as the dawn phenomenon (as above liver releasing glucose).
You don't say how often you have your background insulin to try to control your night time levels when you quick acting insulin is no longer having an effect on your BG.
For me I switched from 1 basal jab a day (40u levemir) to 2 as Levemir doesn't last 24 hours (24u at 6am and delayed the evening one (17u) until 10pm which helped minimise dawn phenomenon and kept levels stable through the night after my bolus (Fiasp) had worn off (5 hours)
I was on Novorapid too but the makers added a vitamin to it and released it as Fiasp so it works faster (claims from 2 mins) which is why I asked to switch to it.
Obviously seek your diabetic teams advice before changing anything as I don't know what background insulins you're taking or how often.
Also worth doing a DAFNE course where you will learn about insulins and how long yours are effective for and how to work out how and which ones to adjust and when to keep control including through the night and a whole lot more (carb counting ,sick day rules, etc etc.)
I find an extra 4u in the morning on waking corrects my dawn phenomenon rise.
Good luck