It seems like I fainted while I was running, and by the time I came to, I was in the hospital.
It looks like I passed out jogging, and by the time I realized it, I had already arrived at the hospital.
この場合、次のような言い方ができます。
ーIt seems like I fainted while I was running, and by the time I came to, I was in the hospital.
「私は走っている時に気を失ったようで、気がついたら病院にいた」
to faint で「気を失う」
to come to で「意識を回復する」
ーIt looks like I passed out jogging, and by the time I realized it, I had already arrived at the hospital.
「ジョギンしてたら気を失ったようで、気がついたら病院に着いていた」
to pass out で「気を失う」
to realize で「気が付く」
ご参考まで！
It seems that while I was out jogging, I fainted out of nowhere and when I woke up, I was in the hospital.
ご質問ありがとうございます。
「走っている間に気を失ったようで気づいたら[病院](https://eikaiwa.dmm.com/uknow/questions/32103/)にいた」という場面を説明したいなら、英語で「It seems that while I was out jogging, I fainted out of nowhere and when I woke up, I was in the hospital.」と言えます。
上記の英文の「I woke up」は大丈夫と思いますが、気を失ったなので、「I regained consciousness」の方が適切と思います。
ですので、「It seems that while I was out jogging, I fainted out of nowhere and when I regained consciousness, I was in the hospital.」でも言えます。
ご参考になれば幸いです。
カナダ 
アメリカ合衆国 
