IIM

Yes, you can appear for the Common Admission Test (CAT) again after joining any of the Indian Institutes of Management (IIMs), whether they are new or old. If you take the CAT and achieve a better score, you can apply for admission to other IIMs or business schools of your choice.

However, keep the following points in mind:

Withdrawal Policies: Check the withdrawal policies of the IIM you join. Some institutions may have specific rules regarding leaving the programme after enrollment.

Time Commitment: Consider the time and effort involved in preparing for the CAT while managing your current studies.

Application Process: You will need to go through the entire application process again, which includes submitting your CAT scores, personal statements, and attending interviews for the new institutions.

Career Impact: Think about the potential impact on your career and professional network, especially if you leave an IIM to pursue another program.

A candidate who scores just 20 on the percentile scale in CAT, hence, will find it difficult to get admission for MBA programs to any top part of the elite business schools, for all these institutes have put a very high cut-off percentage on a general basis in the range of 85 to 99 percentiles.

This very high cut-off range for all potential candidates includes even new and baby IIMs. All of them set the bar high even for the reserved categories. But then there remain some options for you.

A few private business schools and deemed universities offer admissions at the lower-end CAT score or even conduct their entrance tests. Many such institutes are Amity University, Lovely Professional University (LPU), and many others that assign those low percentiles additional weightage on other selection criteria, like their academic performance, group discussions, and personal interviews. Furthermore, there are some state universities or tier 2 and tier 3 MBA colleges that offer admission on a no-CAT basis. 

It all depends upon which IIM you get shortlisted, but considering your percentile in CAT 2023 that is 70, a very good chance stands by your side if the cutoff will be met by you in terms of that IIM.
For example, IIM Ahmedabad will set the minimum overall cutoff at 60 percentile for ST candidates while others may be different. Normally, top IIMs such as IIM Ahmedabad, Bangalore, and Calcutta are higher with cut-off values generally more than 90 percentile in most of the cases but relax it for the reserved categories.
Although your score may suffice for admission into some IIMs, admission is not guaranteed and will depend on your overall profile, including academic performance and performance in subsequent rounds like interviews.