Mostly cards issued by bank comes pre-activated with ATM and offline POS txns enabled. Not the case with CC though which requires a txn in 30 days or else deactivates. So you better hotlist it only after receiving it is only thing I suggest to save ESP from getting deactivated.
How system works at HDFC.
Current Card - Request for new Card - Card Delivery Confirmed to Them by partner courier - System will deactivate old card and keep new active in next 3-5 days.
But if system don't find two active cards after you hotlist Millennia, it wont close your old card.
Also even if you keep it idle I can't confirm 100% whether they will close ESP or not. In past they didn't closed older one till new one wasn't activated (Setting PIN and enabling ECOM). Setting PIN itself is a mode of activation for new card. That's why I asked you not to set pin or even do anything to it.
For my own changed card I activated after 17 days and after that old one closed after 4 days of it. But not sure whether same thing is getting followed now or not.