You can't use the exclusion more than once every two years, unless the reason you are selling the second home is due to one of the types of unexpected financial hardships as described here, under "do I qualify for a partial exclusion."
https://www.irs.gov/forms-pubs/about-publication-523
Assuming you don't qualify for a partial exclusion, this is how it works.
You can only have one main home at a time. If you moved into house #2 in May 2020, then house #1 stopped being your main home at that time.
You qualify now for the exclusion on house #2, since you have owned it for at least 2 years and lived in it at least 2 years as your main home. However, if you used the exclusion on house #1 that you sold in September 2020, you can't use the exclusion again until September 2022 (2 years and 1 day after the previous closing date, to be exact.)
If you want to sell house #2 before September, you have two options.
1. Don't claim the exclusion, and pay the capital gains tax.
2. File an amended 2020 return to pay the capital gains tax on the sale of house #1. By not claiming the exclusion on house #1, you can claim the exclusion on house #2 at any time after May 2022. (Because the tax on 20K of gains is less than the tax on 60K of gains.)
If you wait until September to sell house #2, you can claim the exclusion on house #2 without changing the exclusion on house #1.
If you DO qualify for a partial exclusion due to unforeseen hardship, it will be based on the shortest of three time periods
- how long you owned house #2
- how long you lived in house #2 as your main home
- how long since you last used the exclusion
In your case, (c) is the shortest, about 21 months at this point, so you would qualify for a maximum exclusion of $250,000 x 21/24ths. That would still more than cover your $60K of gains.
