I am about to purchase an ASP and explore, BUT

[Apocryphic]

A hauler is fine, you'll have no trouble exploring within a few thousand light years.

However, if you're going out on a long journey, the extra range really helps. The same with an A6 fuel scoop to reduce time spent refueling. You don't have to get the A6, though, it does cost twice as much as the rest of an exploration Asp fit. Even a D6 is a huge improvement over the A3 scoop you'd have in a hauler. If you're in a hurry to get somewhere, such as when departing from and returning to known space, you can do about 2kly/hr. That's a big difference if you're shooting for the core to visit A*, or any other journey of tens of thousands of light years.

Remember that the galactic disk is only 1-2 thousand light years thick. Changing to a different elevation on the z-axis can help.

This is a great point, too. There's a lot of volume to explore, and you'll start finding numerous collapsed stars at coordinates further than 1kly in each axis from Sol. The disc thickness has an even greater variability, though, from more than 5kly at the galactic bar to less than 1kly around the outer voids. This is roughly in line with the apparent density on the galaxy map.

 

[Hellhawk666]

Don't plot a course straight out to an iconic destination. I guarantee if you plot a course from Sol to Maia you'll hit nothing but already discovered systems. Try aiming for somewhere 100LY away from your intended destination, but at the same distance, so you're not following a well-beaten path. Sometimes you only need to detour a tiny amount to hit heaps of unexplored systems. Plot a course to somewhere of no interest at all and once you're past the "casual explorer" region nearly everything will be unexplored. The iconic stuff (nebulae stars, big stars, named stuff, etc) is getting snapped up quickly. It's hard to find something important that is unexplored within 20k LY now, but there is masses of "small stuff" - enough for years to come