See there are no such restrictions for the improvement examination to take admission in the 11th without the 10th improvement examination done. But there can be some school which may not allow your admission but as per the internet, most school allows the student to get admitted.

So I would recommend you to contact the schools in which you want to take admission and get it discussed, they will give you all the valid information required.

For CBSE improvement examination, visit the link below:

https://news.careers360.com/cbse-improvement-exams-2021-check-class-10-12-date-sheet-eligibility-criteria-here

I hope this answer helps. All the very best for your future endeavors!