Increasing the headroom of Opamps usually allows you to increase the signal to noise ratio by increasing the gain and thus signal amplitude. Also it helps avoiding clipping due to output limitations of the operational amplifiers.
The +-5V Dual supply will increase the headroom compared to a 5V single supply, but so will a +10V single supply. With dual supplies you don't need virtual grounds and less/no coupling capacitors between stages.
The supplies and the Opamps have to match. Not every Opamp is designed for dual rails or a total supply voltage of +10V.
For low current requirements it's easier and likely more efficient to use a charge pump. Check the current consumption of your output stage and whatever load/receiver you want to connect it to. If you only use a couple of Milliamps a charge pump would be a good choice.
However the switching frequency of many charge pumps ICs are within the audio frequency band or close to it. If the switched capacitor is a multilayer ceramic capacitor (MLCC), it can start to vibrate mechanically and audibly due to the switching.
Using a higher switching frequency or a different capacitor type avoids this.
A higher switching frequency should also reduce the ripple of your supply.
But power supply rejection ratio (PSRR) goes down with increasing frequency, thus decreasing the former effect.
You can also put a low-dropout linear voltage regulator after your switching inverting stage to get a cleaner negative supply voltage.
If you want to learn more about Audio quality in analog stages I recommend reading "Small Signal Audio Design" by Douglas Self. The chapter about power supplies only uses line voltage transformers as first stage though, no switching supplies.