Some years ago, I found an 1944 "Education Manual EM910 - Elements of Aeronautics" for the US Armed forces in a used book store (with the original paper compass in tact!). While it covers detailed math on topics of airpseed and such, I did find an interesting estimation tool within. On the topic of how airspeed is measured
From Part I, Chapter 4 "Basic instruments used in flying", Pages 35-36
"Air-speed indicators are graduated (scaled) to indicate the speed of the airplane at sea level at normal temperature (16deg C). But at higher altitudes, where the air is less dense, the difference in pressure between the two tubes* is less for a given air speed than it is at sea level. Hense the air speed indicated at altitudes above sea level is less than the true air speed. (A means of determining altitude is described below.)
The reduction in indicated air speed is about 1.5 per cent for each 1000 feet. Hense to find the true air speed an approximate correction can be made to the indicated air speed by adding 1.5 per cent to the indicated air speed for each 1000 feet of latitude. Thus, if at 6000 feet the air speed indicated in 100 miles an hour, the correction is 6 X 1.5 per cent, or 9 per cent. Nine per cent of 100 miles is 9 miles. Hense the true air speed is about 109 miles an hour.
Roughly, the rule works the other way also. That is, if the true air speed at 6000 feet is 100 miles an hour, the indicated air speed is about 9 per cent less, or about 91 miles an hour."
While this is an estimation, I find it fairly accurate for the GA planes at lower air speeds. You would want to cross test it against the GPS to see what it's practical ranges are, and then could use it to provide an corrected IAS at altitude to use with ded reconing. The chart in the book shows a graph from - to 20,000 feet, with speeds from 40 to 160 mph.
Many other sources show the multiplier to be from 1.5% to 2.0% per 1,000 feet. Given that we can only do very rough approximations with this rule of thumb, any of the above values may be used. For true accuracy, the indicated airspeed (IAS) must be corrected for barometric pressure, altitude and temperature to get true airspeed (TAS). Pilots and navigators generally do this with an E6B flight computer. While the circular slide rule version of the E6B has been around since WW II or before, in recent years electronic versions have become available. In fact, the Sporty's electronic version, when calculating TAS, asks first for pressure altitude, then for temperature in degrees C, then for calibrated airspeed (CAS). It then gives TAS, mach number and density altitude as results, somewhat beyond the normal outputs on the circular slide rule version, though you can read a different scale on it to get density altitude.
You could also figure the thumb rule for TAS as around 8% for every 5,000 feet.
*Note that the two tubes refer to the pitot tube and the static port. The static port
on some aircraft is actually a separate tube near the pitot tube. On many other aircraft, the
static port is located on the side of the fuselage or at another location that the design
engineers have found to be minimally disturbed by the aircraft's forward motion.
To compare to the rule of thumb, the list below shows the true airspeed actually calculated with the Sporty's electronic E6B for standard conditions (29.92" Hg, 1? C or 5? F, ? C lapse rate) at a calibrated airspeed of 100 knots:
Sea Level -- 99.8
1,000 ft -- 101.3
2,000 ft -- 102.8
3,000 ft -- 104.3
4,000 ft -- 105.8
5,000 ft -- 107.4
10,000 ft -- 115.9
15,000 ft -- 125.3
20,000 ft -- 135.9
30,000 ft -- 161.1
40,000 ft -- 193.3
Note that the same exact values above work whether you use knots, miles per hour or kilometers per hour as your units, so long as the units used are consistent throughout.