The current .380 craze didn't start with the LCP. Perhaps you are thinking of the Keltec P3AT that the LCP was copied from!
The .380 has always been a popular S/D cartridge, and there really is no SAAMI "+P" designation. Any good hollow point round should work well. Caliber is not a replacement for accuracy, but you should feel confident in what you are carrying. A bad shot with a .380 will still be a bad shot with a 9mm.