score:6
Looking at the available flights, you have two options:
Flying east you could stop in Europe, common stop-overs are London, Paris, Amsterdam, Frankfurt, Vienna, or all the way to Dubai.
When flying west in addition to Singapore and Hong Kong other options are: Seoul, Bangkok, Tokyo or Shanghai.
All these cities are interesting enough for a few days or more (I haven't been to Seoul, so I can't judge it).
Next check 'Visa'. Which of these countries can you visit without an advanced visa. Of course this depends on your nationality. I assume you have an Indian passport. I don't think a US Green Card helps.
A good source for the question 'Do I need a visa' is VisaHQ
It rules out all the European cities, also the Emirates. Of all the Asian stop-overs only Hong Kong is listed as visa free.
You can visit many Island states in the Caribbean and Pacific but none of these 'free' countries are really helpful as a stop-over.
So Hong Kong it is!