Use parts with lower temperature coefficients (NP0 or C0G for ceramic capacitors for examples). This is typically the most expensive option, but is simplest at a first-pass optimization.

Use a quality voltage regulator to supply the oscillator, one that is immune to temperature variability within the design's operating temperature range.

Minimize the dependency on variable components (capacitor, inductor, or resister) for tuning. Pad the variable components with fixed components to minimal the variables component values. For example replace a 100k$\Omega$ potentiometer with a 47k resisters on the two legs of a 10k$\Omega$ potentiometer as potentiometer may have a temperature coefficient of 1000ppm, while the fixed metal film 1% resistors may have tempco of 200-500ppm.

Use parts with complementary temperature coefficients of components (or additional components with a temperature variability that is well characterized like a thermistor e.g. +10 ohms per degree increase) that cancel out a change in values when the temperature changes. Eg. Oscillator drift correction circuit (mainly in reference to RF oscillator drift, but the principles are consistent)

Use parts with lower temperature coefficients (NP0 or C0G for ceramic capacitors for examples). This is typically the most expensive option, but is simplest at a first-pass optimization.

Use a quality voltage regulator to supply the oscillator, one that is immune to temperature variability within the design's operating temperature range.

Minimize the dependency on variable components (capacitor, inductor, or resister) for tuning. Pad the variable components with fixed components to minimal the variables component values. For example replace a 100k\$\Omega\$ potentiometer with a 47k resisters on the two legs of a 10k\$\Omega\$ potentiometer as potentiometer may have a temperature coefficient of 1000ppm, while the fixed metal film 1% resistors may have tempco of 200-500ppm.

Use parts with complementary temperature coefficients of components (or additional components with a temperature variability that is well characterized like a thermistor e.g. +10 ohms per degree increase) that cancel out a change in values when the temperature changes. Eg. Oscillator drift correction circuit (mainly in reference to RF oscillator drift, but the principles are consistent)

You could consider a low frequency crystal oscillator such as the 32.768 kHz commonly referred to as a watch crystal as it is typically used in Real-Time Clock (RTC) circuits, as well as low-power microcontrollers. Using it in a VXO with a small tuneability, (aka "pull") ~10% I believe and a frequency divider you can generate a very stable audio oscillator that is tuneable over a narrow range.

The other is to make the oscillator circuit's environment thermal stable using a) insulation to minimize and slow thermal change, and if necessary b) temperature stabilized heat / cooling such as an oven-controlled crystal oscillator, OCXO.

If you look at quartz oscillator modules or "cans", pay attention to their output type, most are designed for digital timekeeping / clock generation and only output a digital signal, though sine-wave or clipped sine-wave modules or XO are available.

In response to necessary stability, that is application dependent. If you want to be able to match the oscillators frequency over time (such as in a multiple track recording studio, where tracks are layered and each track is recorded separately) then stability is important because while absolute accuracy of human hearing is moderate (no better than 1% I would guess), relative frequency mismatch is easily detectable to a much smaller degree (again I would guess approx 0.1 - 0.01%).