If a lower voltage is acceptable a low quiescent current & reasonable efficiency buck regulator will help. If as you suspect, Ioperate is constant with falling voltage then time duration will increase as 1/V. I lower V is provided by a buck converter with efficiency Z ( 0 < Z < 1).
Typically Z = 0.8 to 0.9)
Then new duration ~= Old duration x 12/Vnew x Z
eg For Z = 0.85 and Voperate = 8.5V then
Tnew = Told x 12/8.5 x 0.85 = 1.2
= 20% gain.
The maximum possible gain is at 8V (JUST operate) and 100% efficiency (not achievable)
= 12/8 x 1 = 1.5 = 50% increase.
It MAY be possible to operate the device with a 12V pulse of lowered net duration - whether this is possible depends on aspects that you have not told us about. A proper description of the problem and equipment is needed. It is possible that a MUCH longer life is possible but we need complete information.
Current drain is expressed in amp(ere)s (A)
A x H = Ah.
G_G's Joule thief is well intentioned BUT as a general rule, most circuits with that name are built by people who are not aware of what they are doing and the results are not usually marvellous. (The Wikipedia generic example is a typically sub-optimum one). This does not HAVE to be the case, but often is. A modern (or ancient) buck regulator OC (or discreet circuit properly designed will usually be vastly superior.
Here is an example from Elliot Sound Products, of a discrete component buck converter that works well and constant current version here
I know it works well because: Here is a remarkably similar design based on a circuit provided to me by God in 2001 (but, that's another story* :-)). This uses a PFET as the high side switch but this can equally be a small bipolar transistor in this case. The design is remarkably similar to the ESP circuit. As mine appeared on the web some while before his he may have copied the GSR (which anyone was and is freely allowed to do) or he may have received inspiration from another source. It's a "hysteretic converter - which was a well enough known idea at the time - but one I'd not until then heard of. (Despite some extremely smart people saying it has no hysteresis, it does have. The ripple on CBUK2 provides hysteresis and is required for operation. The output is "chaotic" and spreads the EMI signature nicely. For Vin < Vout_design is is well behaved with high side switch on. Switching starts seamlessly as Vout reaches design point.
*. Note specific time and date on circuit - to nearest minute :-). Ask off-list if of interest. An unknown number of these devices were built thousands and possibly 10's of thousands, and they filled a need very well. [I was meant to get royalties - this was a small part part of a larger design but 'things went wrong with payments' as they do][Several paid visits to Taiwan resulted so it was fun overall.] As used the circuit converted Vin of from 12V to 200V+ down to 12V out. Efficiency in the normal 12V-50V range was acceptably good and while only about 50% efficient at 200 Vin, that is vastly superior to the 12/200 = 6% efficiency of a linear regulator. At eg 5W out energy loss is 5W at 200V with this regulator and 78+ Watts with a linear regulator. Most sources say that a buck regulator with a 200/12 = 16:1 + input voltage range is not practical, but God knows why. [ :-)].