Research and common sense both say that our brain doesn't go to sleep or switch off when we stop doing brain things (thinking, problem solving e.t.c.). And I am sure we all know that. The part most of us hardly consider is that: what then is our brain doing when we are sleeping? That's the question that will form the foundation of all I am going to share with you.
Our brain is constantly working. It is using all the information/education/smartness you have acquired to help you consciously come up with solutions and plans during your waking conscious hours, the same way, too, it is using those very resources to come up with plans and solutions when you are asleep (or at your unconscious mind level). The only difference is that the plans and solutions it helps us come up with during our conscious moments we claim as ours and are not surprised by them, but when it comes during our unconscious moments we call them "gut feeling" and "intuition".
In essence, the difference between your strategic deliberate plans and your gut feelings is at what moment you received them. One came during your conscious moments and the other maybe when you were not 100% controlling your mind. But the processes and resources used are the same.
Garbage in, garbage out. If you make terrible deliberate plans because you are not well educated or feed yourself inadequate useful information or not smart, then your gut feeling will be a fountain of horrible ideas. But if you improve the quality of resources your mind has access to and build up the capacity of your mind to think critically, your gut feelings will be excellent quality.
If you want excellent and breakthrough-triggering intuition, you should feed your mind with good quality resources and train your mind to think productively deep. And that is how you improve your gut feeling.
All the best!