The smaller the action the more finicky they tend to be; hence, the need to do a thorough break-in. I'd stay away from any hollow point until you have several hundred rounds through the LCP.
And on the subject of hollow point ammo....I am in the camp of people who believe that .380 hollow point isn't the best choice for SD purposes. The .380 is a marginal SD caliber at best, and particularly in a situation in which you might need to penetrate heavy clothing the hollow point round has proven to be much less effective from the standpoint of penetration.
I recommend shooting a good, reliable FMJ round. Some folks go to the trouble of using a hollow point during the warmer months, i.e., less clothing, and the FMJ for winter.
Your gun, your rules.
XD-40 service, XD-9sc, member GeorgiaCarry.Org, National Rifle Association, Gun Owners of America, North American Hunting Club, Rocky Mountain Elk Foundation