As a law enforcement officer I daily walked around with a pistol in my hand. I was usually confronting felons or I was actively searching for a felon.
As a civilian I try very hard not to search for bad guys. I have very little authority in the written law to capture bad guys. That authority is further lessened by case-law. Usually if I need to search for someone as a civilian there is some extreme circumstance and it usually involves inside my home, i.e. someone in my home with my family present.
Under the few circumstances that I want a pistol in my hand but have not intended to immediately use it you must decide how to move with the pistol.
I use the number two position of the draw where the pistol is parallel to the ground and pointed in front of me. I have two hands on the pistol and it is close to my body. Holding it close to the body keeps the pistol from obscuring my line of vision and keeps any potential bad guys from easily grabbing my pistol. I can fire the weapon from this position as a retention position and can modify it into a true retention position very quickly.
Many people are fearful of carrying a pistol with it pointed forward and parallel to the ground. They instead prefer to tilt the muzzle down at a 45 degree angle in-case of a negligent discharge. If you prefer this that is fine. Just remember that there is no safe direction to point the pistol in the real world. If you live on a second story then you are pointing the pistol towards someone. Don’t feel that where you point the muzzle alleviates you of being safe with regard to when you put your finger on the trigger (i.e. don’t walk around with your finger on the trigger, no matter how you decide to hold it.)
If you find the bad guy and need to shoot it is just a matter of finishing the presentation you have worked on in dryfire practice.
I will close with a hotly debated topic of holding the pistol down at your side but hidden behind your leg. For law enforcement officers there are times that you might do this. A misunderstanding is that this makes you faster. It does not. As a civilian having a gun in your hand and not pointing it can very quickly turn into reckless display, disorderly conduct or a few other felonies. I try not to say never but if your understanding of the law is solid you will usually have the pistol in the holster, pointed at someone, at the ready if you are looking for a bad guy or shooting someone. Please, please study the laws of your state and become an expert at it. Do this before you become an expert at your pistol. Study Wisely.