Is there anything ordinary that melts between about 500 and 590 Celsius? Everything I find is either well below or well above

The gap is real for pure elements and not real at all for alloys, and that distinction is the answer.

Among common metals, the pure melting points genuinely cluster either side of your window — the low-melting ones are well below and the structural ones are well above. There are elements in the range but they are not things you have lying around.

Alloys fill the gap completely, because alloying almost always lowers the melting point relative to the higher-melting component, and the amount depends continuously on composition. Aluminium alloys, magnesium alloys and various brasses and bronzes cover the 500 to 600 band densely.

So the question to ask is not "which material melts at 550" but "which alloy composition melts at 550", and there the answer is that several do and the composition is a dial rather than a fixed value.

For a temperature reference specifically, I would push you towards the purpose-made products rather than hunting for a material.

Welders and heat treaters use temperature-indicating sticks — they look like a crayon, you mark the workpiece, and the mark liquefies within a few degrees of a stated temperature. They are made in closely spaced increments right through your range and they are cheap.

There are also indicating lacquers and self-adhesive labels that change colour irreversibly at a rated temperature, which are better if you need a record afterwards rather than a real-time signal.

All of these are formulated mixtures chosen for a precise melting point, which is exactly the thing that is hard to find among ordinary single substances. Somebody already solved the problem you are solving.

One caution if you go the alloy route and it matters to you: most alloys do not have a melting point, they have a melting range.

They start to soften at one temperature and are fully liquid at a higher one, with a mushy zone between. For a temperature reference that ambiguity may be unacceptable — you would see gradual slumping rather than a clean transition.

The exception is a eutectic composition, which melts sharply at a single temperature like a pure substance does. If you want a crisp indication, look specifically for a eutectic in your range rather than any alloy that happens to be listed with the right number.

There are also non-metals sitting in that band worth knowing about — several common inorganic salts melt in the 500 to 700 region, and some hydrated ones lower.

Whether any is available at home depends entirely on what your home contains. But if you have access to a lab supplier rather than only a hardware shop, the gap stops looking like a gap. It is an artefact of restricting yourself to materials that turn up in everyday objects, and everyday objects have no reason to sample that temperature range.