I don't think any decent SP or agency is going to offer a lower rate, when they can make more $ in private. As an add-on, MAYBE. By itself? No way.
There are SP's that might add in extra time, but you're paying on top of their normal minimum rate (e.g. 3-hour "social", 1 hour public, 2 hr private, say $500, instead of the 2-hour incall). In theory, you're paying $100 for that 1st hour, but overall they are getting 500 total. Most of the time, honestly, it's a small discount, not half-price (if you assume 200/hr is the going rate).
The only SP's that would charge 100/HR for social are charging close to that for private - and I don't know if anyone wants to go that low talent-wise.
Plus, for many SP's, privacy is a priority.