Right away; without delay — emphasizing doing something as soon as possible.
問題があれば、すぐにでも連絡してください。
If there's a problem, please contact me right away.
その書類はすぐにでも提出したほうがいい。
You should submit that document as soon as possible.
具合が悪ければ、すぐにでも病院へ行きなさい。
If you feel unwell, go to the hospital immediately.