Having built some 30 of the Oatbox Radios from the XTAL Set Society, most with my physics students, I decided to see if I could jazz it up a bit in terms of selectivity without sacrificing too much in the way of sensitivity.   I also wanted to add as little as possible to the cost.  The drawing of the finished product is shown below:

  

Construction details:
        The coil used in my prototype version was 60 feet of plastic coated #24 stranded copper wire, wound on a recycled 40 oz plastic peanut butter jar, tapped every 5 to 10 turns.  For the variable capacitor, I used a mica rf tuning cap from Mouser, and mounted it in the lid of the jar.  The main coil is 50 turns, and another 9 turns were close wound over one end of the main coil to be used as the antenna coil.  An alligator clip from the diode is used to pick the best tap for selectivity and sensitivity, which will vary depending on time of day and propagation conditions.  If you are starting with a basic oatbox radio, just add on the extra turns needed for the antenna coil.  You might even try putting them alongside the tuning coil, wrapped over a couple of layers of paper so that you can move it away from the tuning coil and maybe buy yourself a bit more selectivity - I wound mine the way I did because I was running out of level space on the jar.

Testing and reconstruction:
        Initially, I merely added the variable capacitor, and that worked pretty well by itself.  Next, the antenna coil was added, but only inductively coupled to the tuning coil.  Measurements of voltage across the 47k resistor with a digital multimeter indicated about a 30% increase in signal strength by connecting the tuning coil to ground as shown.  Finally, I checked the frequency range of the set, and was able to tune from about 750 to 1600 khz, getting my four local stations at 790, 1100, 1440, and 1490; by local I mean within 20 miles.  There was a strong (50 kw) station about 80 miles away that I wanted to get, so I added the 100 pf fixed capacitor, which I could switch in for low range, extending down to about 600 kHz.
      Yes, I know, I could have gone with a standard 365 pf varicap to begin with, but at 10 -12 bucks a pop, felt this was a better way to go, economically, for my physics students.  Incidentally, the variables on the Radio Shack kits come in at about 150 or so pf;  I would like to find a supply of the mica variables at around 360 pf - any ideas out there?
        At night, the long range MW stations come in, and between the hours of 6 pm and 8 am I can always hear someone speaking Spanish, propably due in large to my antenna orientation (N/S).  Sensitivity of the set is that of the basic oatbox radio, and the capacitor seems to be the major contributor to selectivity, as expected, while also lowering the tuning range.  As shown, the 1440 and 1490 stations bleed over to each other, and can only be separated by ungrounding the tuning coil and moving the antenna coil a bit away from it.  Since this one was built for my niece in VA, and her QTH is more benign than mine (doesn't have two stations close in location and frequency, I decided to leave well enough alone.

    No good deed goes unpunished.  I took this baby to VA, and it turns out that the peanut whistle local station, appropriately licensed as WAMM, or something like that,  is close enough to be a real bandmaster, and sits right in the middle of the band.  I was picking it up before I even strung up the antenna.   Next trip up north I'll try disconnecting the ground connection to the tuning coil, pulling the antenna coil away from the main coil to give her some loose coupling.  The 30 percent increase in sensitivity, less than 3 dB, is probably at the expense of a lot more selectivity.  Back to the test bench.  Maybe I'll even take a wave trap with me.  Fortunately, the bandmaster plays contemporary rock, which suits her just fine;  perhaps I should just take a small speaker and a transformer instead.
    For some more flexibility, set this up as the Peanut Special II.