Dear Aspirant,

For the percentile of 96.6 your rank should be around 48K for the safe side you need to come under 40k, you can also check your rank from this site

If you belongs from the northern India then there is a chance that you will get any NIT in this range but you will not get any average or top NIT for the score, better if you take part in remaining 2 attempts of JEE to score better. Here are the some lower NIT's you can consider-

1. NIT Srinagar

2. NIT Mizoram

3. NIT Nagaland

4. NIT Manipur

5. NIT Arunachal Pradesh

etc.

You can also consider NIT Rourkela for dual degree program. To know the best college for yourself you go through with this link according to your rank.

Best Of Luck!!!