I have done a free fix a few times to solenoids. I opened it up, reversed the heavy lugs (if eroded), then flipped the contact disk over to the opposite (copper) side. The possiblility and success of opening a solenoid varies with the design. The newer, mini starters may not allow this.
Check the headlights for excessive dimming when trying to start. If so, your starter is sucking too much juice. If not, it is the switching circuit.
Chances are, given your 'normal' starts, it is the latter.
Also, even a new battery could be faulty, taking only a 'surface charge', as I found.....TWICE...thanks, Optima RedTop. I, twice, replaced starters, added a remore solenoid and had months of headaches, as I thought the (2) new 'wonderful' RedTops couldn't be the problem, could they? After a VAT test showed my rewiring, he starter and the charging circuit to be fine. A new Intgerstate battery has fixed my problem.