What are you doing the nights before?
My recent 6 month lower back ordeal was the result of stepping up my training on the bike without doing the necessary strength work e.g. was hammering away on the turbo but because glutes were weak my lower back was having to do a lot more work than it should, resulting in low back pain, stiffness and eventual spasm and even a herniated disc! Always noticed the stiffness most in the mornings.
I’m now very much of the belief that your the whole hip/glute complex does so much to protect your lower back, even my physio seems quite disdainful of standard ab and lower back strength work, think her general theory is that these muscles need only be strong enough to keep you upright, and that the glutes should be isolating your lower back from most of your pedalling forces etc.