If you're literally 3.6 miles from the airport that should be more than adequate range to pick up literally everything from the airbase itself, even ground traffic between aircraft and the tower, or ground crews as well. Having said that the obvious answer is YES a better antenna will improve things, sometimes rather dramatically, over any stock rubber ducky on any scanner ever made, that's just a simple fact that cannot even be debated. Where I'm located in Las Vegas I'm basically at a mid-point between Nellis AFB and McCarran International - Nellis is about 6 miles NE of me and McCarran is about 6 miles SE of me, and I can get quite a bit of stuff from both without a lot of issues and I'm using what's called an Off-Center Fed Dipole I learned about here at RR, you can find more info here:
http://wiki.radioreference.com/index.php/Homebrewed_Off-Center_Fed_Dipole
Which antenna should you/whoever get is where things get tricky. A discone is a primarily designed to be a wideband antenna overall and while there are some that are designed to specifically operate on the MilAir band (225 MHz to 380-400 MHz) most of the consumer available models do well in the CivAir band (118-136 MHz) as well and looking at the RR database entry for NASJRB it does have quite a bit of traffic in both bands.
I don't think in your situation that a discone will necessarily make a gigantic difference but it can, absolutely, and putting it in the attic could absolutely help things over what you've got right now if all you're using is the stock antenna on the BCD396XT. A telescopic whip would be better than the stock antenna as well but all this depends on your usage: is monitoring the airbase traffic the only thing you actually care about doing or do you actively monitor other bands, public service, Ham bands, commercial traffic (meaning local businesses, etc) which is a question that needs to be addressed.
Honestly, and I'm being completely serious here, I'd say build one yourself - make a 1/4 wave ground plane from coat hangers and an SO-239 chassis mount and be done with it. You'd be amazed at what's possible and considering the pricing on those antennas you linked to I think you'd be better off with something that doesn't cost anywhere near that much (hardly anything really, a couple of dollars at most) or the need to get it mounted someplace proper. Sure you could put one of them together and place it in your attic but again, that seems like a considerable expense for such antennas when something much simpler could do the task just as well if not better in most situations.
Here's the most simplistic set of plans for a 1/4 wave ground plane you're probably going to find:
For a 1/4 wave ground plane designed to handle the CivAir and MilAir bands you'd be looking at the main vertical element (A) being 22.1 inches (cut to 127 MHz which is the center of the CivAir band) and the 4 ground elements (B&C and D&E) being about 23-24 inches - it's usually a good idea to give the ground elements just a tad more length so about an inch or so can't hurt there but you can make them the exact same length if you wish, it's not going to ruin things to any measurable degrees. The mount is an SO-239 chassis mount like this, available for a few bucks from Fry's or maybe another electronics store near you or available on Amazon/eBay/etc:
You can attach the elements directly (assuming it's coat hangers but you can use anything that's similar like #12-14-16 gauge household electrical wiring, etc) by soldering them or using spade lugs with screws and nuts for the ground elements, soldering the main vertical element in place, there's many ways of physical constructing one.
The good thing is that because of how it's designed it'll actually receive the MilAir band
fairly well at the same time - your close proximity to the airbase itself ensures there's no way things aren't going to be heard when you get not only a better antenna like this one you can make in 10-20 minutes at most once you have the parts together but even more so if you can get this antenna up in your attic. It doesn't require being mounted on a pole, specifically, it can just be set someplace in the attic where it's at a higher point than anything else. If needed you could even do this: add another inch or so to the length of the main vertical element and twist it into a loop to be used for hanging this 1/4 wave ground plane from a hook screw you put into a beam someplace in your attic so the antenna won't take up any direct space on the flooring of the attic.
Many possibilities to do this very cheaply and with vastly superior reception potential over the stock antenna and also saving you a ton of cash as well.
But that's just my suggestion - I'm not trying to take money away from the antenna manufacturers or the resellers of their products, it's just that sometimes paying such amounts of cash for something you can literally make yourself for a few bucks and be proud of it when it proves itself to be very useful is more rewarding to me I suppose. That's part of this hobby and doing it yourself more than anything else.
As for the mobile situation, I'm sure that antenna would be a nice choice but others will chime in on that soon enough, I don't even technically own a scanner anymore myself, I've been using SDR hardware and software for several years now but I monitor CivAir and MilAir just fine off my laptop.
