As others have noted, the big issue is getting the resistance of the wire into the range where your multimeter can measure it accurately. For that, the simple approach is to make the wires as long and thin as you can in order to maximize resistance.
That said, another thing you can do is improve the precision of your measurements, e.g. by using four-terminal sensing, a.k.a. Kelvin resistance measurement. For this, you'll need to pass a current through the wire and measure both the current and the voltage drop across it:
Image source: All About Circuits vol. I, chapter 8.9
This arrangement allows you to exclude any additional sources of resistance along the current path, such as the contacts between the voltage source terminals and the wire, from the resistance measurement.
Note that, while the circuit shown above includes a separate voltmeter and ammeter, you could also replace the ammeter with a shunt resistor with a known resistance and measure the voltage across it, as in the circuits shown at the bottom of the page linked above. This would allow you to do the measurement using only a single voltmeter. As a bonus, the shunt resistor would also serve to limit the current across the circuit.
Warning: Never connect a voltage source, such as a battery or a simple lab power supply, directly across a low-resistance wire. This will create a short circuit, potentially causing the wire or the power supply to overheat. Instead, always include an appropriately sized resistor in series with the power supply to keep the current down to a reasonable level.
For an even more accurate resistance measurement, you could set up a bridge circuit, like the basic Wheatstone bridge shown here:
Image source: All About Circuits vol. I, chapter 8.10
Such circuits can allow very accurate resistance measurements by comparing the resistance to be measured with resistors of known values. In particular, for measuring low resistances, you might want to look at the Kelvin Double Bridge circuit described further down the linked page.