24/7 IT Support for MSPs: How to compete without night shift staff?

My clients expect support at 2am but I can't afford a night shift team. I lost a client last month because their "emergency" (forgot password) at 11pm wasn't resolved until morning. They switched to a competitor who has 24/7 coverage. The client had 8 users. We are talking maybe $800 MRR but the math just doesn't work for 24/7 coverage at that scale even if I hired someone part time for nights, I would need to charge way more than the market will bear for small businesses and most of these “emergencies” aren't even real emergencies. It's password resets, "how do I access the VPN," or "can you install this software." Stuff that could totally wait until morning, but clients don't see it that way. I have thought about hiring offshore support but I am worried about quality and communication issues. I have looked into answering services but they can't actually solve technical problems. How are other MSPs handling this gap between after hours expectations and what's actually sustainable at smaller MRR clients?