You don't have to book separate tickets. Use Google Flights and book a multi city. Much easier and keeps it on one reservation. And many airlines DO allow stopovers up to 5 days even to encourage tourism. For instance, Korean Air.
My friend was looking at a ticket Charleston SC - ATL - ICN round trip 1600$
It is CHEAPER for her to fly CHS-ATL-ICN stay in Korea for a week
AND THEN 7 HOUR NONSTOP TO BALI for one week
then back to USA Bali-ICN-ATL-CHS
For 1400$!!!!!!