It's because the time loop works for roughly 24 hours. As Rita said, the Omega would 'reset the day', which meant roughly 1 day prior would be as far back as the Alpha/Cage would go. Cage's day before the invasion goes roughly like this:
- Wakes up in the early morning on helicopter. Meets General, gets tazed.
- A few hours pass.
- Cage wakes up in the mid to late morning. This is where the time loops reset to throughout the film.
- Day of training on the military base.
- Day ends.
The next day, the invasion occurs. During that day, they wake up, get in gear, and ship out; hours would have passed during all that. Then Cage manages to kill the Alpha, and get its blood mixed with his own. This must have happened 24 hours after step 2 above, during the hours that Cage was out. Alternatively, it happened to occur exactly 24 hours after step 3, but the hours of him unconscious provides a nice window where if time is reset he wouldn't be able to alter his fate of being sent on the invasion.
Now, why did he go back to earlier when the Omega was destroyed? I offer these two assumptions:
- Omega blood functions the same way as Alpha blood.
- The time reset is still only 24 hours.
We don't know that 1 is true, as no one in-universe knows this. 2, is able to be true while still enabling what we see in the film. The attack on the Louvre occurs earlier in the day than the invasion. During the attack on the Louvre, it's still dark out. J squad had been recruited for this mission during Step 4 above, the day of training. They then get convinced to go, and head out. They arrive during the evening/early-morning. The mission is a success, and the Omega is defeated. This would have happened 24 hours after a point in time prior to Step 1 above, while he was asleep on the helicopter. It would have triggered the start of a new time loop from that point it time.
Ergo, Cage was sent back 24 hours prior to when the Omega was killed, which means he wakes up in the helicopter.