If you had 39 feet of wire spiraled around a 14 foot long tube I might feel that this would help load it up on the lowest band. The serpentine, I do not think, helps it be electrically longer than 14 feet.

To a first approximation, the useful electrical length (for radiation purposes) of almost any antenna is the physical length - no matter how many times it folds back etc. This assumes that the antenna is essentially 1-dimensional. Extra wire running alongside the other wire can modify the feedpoint impedance: basically you have incorporated a distributed matching network in the antenna, which will add reactance. It may also widen the bandwidth, as you have a thicker antenna.

Once you start adding capacity hats or using wider spacing or more wire going off at right angles then the electrical length can increase, as you are now using 2-dimensions.

I seem to recall seeing an academic paper on a fractal monopole antenna, which showed that the electrical length was not much different from that of a simple straight monopole which extended the same distance away from the ground plane.

EZNEC won't correctly simulate adjacent wires that close together so I placed them 2" apart. It shows two resonances at 8 and 17 MHz. The input impedance at 7 MHz is 3.4 - j112 ohms.

At 7 MHz a single 14' wire presents an input impedance of 4.4 -j700 ohms.

Adding four 2" top hat spokes makes the input impedance 8 - j380 ohms.

Yes, so radiation resistance stays essentially unchanged when folding back occurs but folding adds some inductance which reduces the capacitive reactance of a short monopole. Then adding a capacity hat nearly doubles the radiation resistance by nearly doubling the average current.

You could combine the two features. Maybe 14' out, plus the same back to form an open folded monopole. This is 14' long for radiation purposes, but has added inductance due to the transmission line mode of the pair of wires shorted together at the far end. Then add a capacity hat to increase the radiation resistance and further reduce the inductance.