If you weren't doing this already, try holding your breath after exhaling to reduce the amount of oxygen stored in your lungs, then the wait wouldn't be as long - 2-3 minutes is a good breath hold!
Actually I just tried it and yeah for me at least the full-lungs version is obviously longer but takes less discomfort for my SpO2 to start to drop (like well before contractions start), whereas on exhale I get more discomfort (around 15s before contractions) before a similar drop.