Avoiding compression, but raising level

[steve]

@kozikowski

If you insist on doing it manually, you’re describing Effect > Leveler.

No
The “Leveller” is a “wave-shaping” effect. It shapes (distorts) the waveform and will ruin the sound quality.

To reduce only the peaks that go outside of the +/- 0.5 range, the appropriate effect is a “Limiter”.

Try the “Brick wall limiter” from here: Peak Limiter
I’d suggest that you try the default settings first and see if that gives you enough volume boost.

Instructions for installing this plug-in: Missing features - Audacity Support