The value of using a "rod" that Searcher and Imshooter mentioned can NOT be overestimated.
You can buy 1/4" steel rods at home depot and the like. The one I used years ago was about 3 ft long, and I had a section of pipe welded on the end at a local garage to form a T-handle. Sharpen the end on a grinder (doesn't have to be needle sharp, just "pointy"), and "voila"!
Home dumps were just holes in the ground a few feet deep. They'd toss their trash in them and periodically cover them with a layer of dirt, especially if they were used to dispose of organic material. When the dump got full, they topped it with a final layer of dirt. Then they'd start another one right next to or close to the last one.
Beyond the "well, duh!" issue here, the concept you need to take is that these areas are "disturbed dirt", and have a different "texture" than surrounding dirt.
You can cover a LOT of ground just by pushing your probe a few inches into the ground every couple of feet. This is usually sufficient to find spots for further investigation, but that can trick you if the ground has been historically soft for millennia - in which case the difference in surface texture may be difficult or impossible to sense. But even that will work to your advantage for step 2, which is:
When you find a disturbed texture (or the ground is too soft to find any particular area), you want to probe a foot or 2 deeper into the ground.
Dumps will have everything from bottles to broken dishes to broken plow blades to biodegraded organic items, and these all produce "voids" in the ground surrounding them. I have NEVER come across a dump that didn't feel what I can only describe as "crunchy" as the probe tip passes through them, hitting infinite combinations of soft to hard to hole to pushing debris aside.
These techniques aren't foolproof by any means. I've hit sites that I KNOW DAMNED WELL had to have had a dump or 2 on them and come up dry. It might "not be nice to fool mother nature", but she has no problems with fooling around with us!
But this is often balanced by finding dumps in sections I never would have suspected, so it evens out somewhat.
But of this you can be sure: going dump hunting without a probe is like going out coinshooting without a metal detector.