You don’t have to wait for it to cool. As water evaporates the dissolved solids in it reach a higher concentration. If the concentration becomes greater than the molecule’s solubility it will precipitate and fall to the bottom.
If you have pieces of metal that you can filter out. Metals in water are usually in ionic form, they are "chelated" by water in solutions. Unless some salt is created and precipitate, solved metals distribuite over the solution to avoid concentration gradients.
So the answer is: what metal are you talking about? What is its form and concentration? Most likely, if you couldn't see depositions before boiling it, metal ions will likely stay in solution.
Boiling water is used to kill biological organisms. If you want to get "pure" water you need to distillate it or filter it with materials that can capture ions
They stay in the water when you boil it, what you need is a good filter. Most filters you find in the shop don't do much tbf, but I cannot suggest anything, I am not an expert on commercial products.
If you distillate it you get rid of metals, but you also remove the so called minerals. Distilled water taste bad and is not really good for you
Having done some pilot scale experiments (60 l barrels), I’ve noticed that mixers are absolutely essential. At that scale, metals really do form notable concentration gradients.
solid metals are a separate phase, they create a deposit
salts over a certain concentration, part create a deposit, so they slowly create a powder at the bottom, part stay in solution as ions
Ionic metal in solutions spreads all over, as any concentration difference (gradient) generates an excess of free energy that the system naturally releases. You need to add external energy to maintain the gradient, such as a external electric potential gradient (an anode and a cathode)