I don't have the SIG but I do have the PPK/S. Its a nice pistol for concealed carry because it is flat. In the PPK/S the purpose of the plastic 'thing' you are referring to is to give one more finger a place to grip this small gun.

I have a Sig 232.  It's a great gun.  The PPK would do you well also.  No problems with either.

I would side with the SIG based on better trigger pull.

Just a few thoughts to consider:

First, do you want to bet your life on the 380ACP cartridge?  I don't trust anything less than a 50BMG to stop an attacker definitively.  I carry a 45ACP for CCW and don't trust it all that much.  For me, serious defensive handguns start at 9mm or 38 Special +p.  

Second, why not carry a different pistol in a more serious caliber?  My immediate thought was the Kahr in 9mm.  I just shot one in 40 S&W and was really impressed.  The gun in remarkably small and light weight.  Its size is that of most of the 380's and can be had in 9mm or 40 S&W.  I'm a die hard Sig fan but I was impressed with the weapon's performance.

I like the PPK/S due to it's compactness. It's noticeably smaller than the Sig.  The plastic bases can be filed down for better concealability.
With Corbons, it's a potent companion.
It does tend to bite the web between thumb and forefinger after a few rounds, but it's tolerable.

There are 1911's in 9mm, but they are scarce and tend to be expensive.

I have a Colt Pony Pocketlite in .380 it's DAO and I like it's low weight, low snag ability for an always gun, but, if you want 1911 style controls just get a Mustang.

check out the guys at http://www.colt380.com

Also there are a few 9mm 1911 copies. I have shot a Norinco in 9mm that shot as well or better than hi-power IMHO.

aimsurplus has a Star Model B 1911-style 9mm for $149

Oh yes, I forgot. Colt also made the Govt. Model in .380, a scaled down 1911, almost identical in size to the PPK/S. I got a Pocketlite version of it for my wife.

Why do you not like the double action? Is it because of the trigger pull?  I can tell you that my wife's Kahr P9 has a [b]long[/b] trigger pull.  I feel very safe with her carrying a round in the chamber because of this. It's also very accurate.  The .380 even in Corbon loads has only 220 ft. lbs.  This is pathetic! For one shot stoppage I don't feel confident with anything less than 500 ft. lbs.

I have found that some PPKs are extremely reliable and some are not.  If you buy one, it will probably come with one magazine with the plastic extension and one without it.  Actually the extension is a nice thing.  Recoil is a little sharp, nothing more.  I’ve seen or heard of too many malfunctions with PPK’s to ever carry one myself.

Sig 232’s are IMHO more reliable as a group.  However, they are a little bulky for the cartridge they shoot.  The fat grip tones down the recoil.  I don’t like the heel mounted magazine release – it dramatically slows down your reload.

Both PPK’s and 232’s are accurate, sometimes extremely accurate, handguns.

Do not get a Sig 230.  Decocking it with a round in the chamber using your thumb puts the hammer all the way forward (you might decock like this if you are used to handling 1911’s).  If the hammer is all the way forward and you drop the gun on the hammer, it can go off despite the firing pin safety.  Using the decock lever like you’re supposed to puts the hammer in a half cock notch. The 232 was redesigned to correct this.

Star used to make a nice fairly small 9MM 1911 clone.  I think they also made a full size 9mm, which I’m sure you don’t want.  Star also made a .380 1911 clone.  Llama also made a .380 1911 clone but quality was erratic.  

Colt Mustang or Government Model .380’s are small, handy little guns, though not too accurate.  They’re no longer made but you occasionally see them at gun shows.

Seems to me the best gun for you, despite your dislike of double actions, is probably the polymer framed Kahr K-9.  Your call, however.