For example: "My doctor's bookings were full today, so I had to go early to try and catch a walk-in appointment."
"Walk-in appointment" is not regarded as 予約 ("appointment") in Japan. It will usually be treated as 予約なし診療 (literally "appointment-less visit") or 予約外診療 (literally "outside-appointment visit") in Japan. This often means waiting for a long time in a hospital.
Other options are:
- 当日予約: if you managed to make an appointment over the phone several hours before the visit
- 飛び込み診療: this is the same as 予約なし診療 but sounds negative