From your increasing number of spot readings and the anecdotal observations from others about Alkaline battery starting voltages I don't see anything that tells me that your meter is actually reading out of spec - you're seeing some voltages higher than you expected but others lower or the same as you expected.
If you are really dead set on using a primary cell as a reference rather than a tightly specified 3 terminal semiconductor one then, in a non-Mercury/Cadmium world, you need to get yourself a Silver Oxide watch battery as I said several pages ago! It has the flattest discharge curve of any of the battery types.
All the silver Oxide datasheets say 1.55V with a flat discharge voltage until end of life. The discharge curves all show a small wiggle or drop over the first few hours, but if you load it with the resistance value specified on the relevant datasheet for around 50 hours, then you will be into the flat discharge region.
Just go and look up the datasheets for various manufacturers, Energiser, Renata, Maxell etc. Just type in "Silver Oxide battery datasheet" and they are all just a few clicks away. There really is no point in messing around with any battery technology, Alkaline, Lithium (better than alkaline but nowhere near as flat a Silver Oxide) etc that don't have a ruler flat specified discharge curve.
You really do need to to some basic research for yourself.
Edit: Silver Oxide batteries are used in other things too, calculators, calipers etc. If it starts SR (rather than LR or CR) then it's Silver Oxide. Buy a new one, load it with the resistor compatible with its capacity (datasheet). Leave it 50 - 100 hours, measure it (still under load). Use a holder of some sort - DON'T solder to it (at best you'll damage its characteristics).