Can anyone recommend a precise 11.0592 MHz Crystal Oscillator?

There is no such thing as "complete accuracy" in any physical object.

12 seconds per hour is a lot, however.

I'd readjust the divider circuit for a standard oscillator, such as 10 MHz. There is nothing magic about 11.0592. At 10 MHz you have more variety of oscillators to buy, and you can get temperature stabilized ones, and even temperature controlled ones (in a tiny oven) that will hold 2 PPM, which is 0.0002%

1 sec per day is about 0.001% or 10 PPM

1 sec per week is about 0.0002% or 2 PPM

A crystal oscillator does not have series capacitor. So what you have is a crystal, with your own oscillator. The latter may be the problem.

One solution (if using a microprocessor) is to take the error into account in the software. As this crystal is not a binary multiple of one second I suspect that it is already doing this, so you need to change a few numbers to suit. Better if it is a user setting rather than reprogramming the constants in the software (eprom). The crystal can be adjusted slightly with a variable capacitor trimmer across it - see the application data for the crystal or the oscillator you are using or the uP data sheet. Crystal oscillators have a mystery associated with them, but basically they are specified with a specific load capacitance that includes the input of the oscillator, the wiring etc. If the spec is 18pF, the inputs and wiring might be 10pF, and a small trimmer capacitor makes up the rest. Note the typical crystal if set to the exact frequency is still going to drift due to temperature etc. The one in the Farnell link below is vaguely specified but seems to be 30ppm calibration accuracy at 25C and 30ppm drift between temperature -10 and 60C (around 0.42ppm/degree). Additionally there is aging like 3ppm in the first year. The actual load capacitance affects the frequency too, so this allows minor adjustment. Maybe a 20pF trimmer capacitor? Initial error at 25C is therefore 30ppm + the change due to whatever capacitor loads it. Lets say 100ppm total error. That error would be about 0.36 seconds per hour, so it needs to be adjusted. If it is not temperature corrected you can expect errors up to 0.1 seconds per hour, even if it is exactly adjusted initially. In practice this will be a few seconds a week as the temperature swings around. This is the reason that people use ticks from a GPS receiver or such precise reference. In ordinary digital watches the stable temperature of the wearer might help. Making more accurate oscillators is an involved subject. You can arrange for software adjustment, and adjust automatically as the temperature changes.

If the division ratio is changed by just one count per tick, the change is approx 1:11million, so 0.325ms per hour. Even fractional changes can be achieved, by changing the count every xx seconds.

Specific frequency crystals are expensive, and not going to resolve all issues anyway, so I don't recommend getting one made to suit. However Farnell in the UK have such a thing. These can be bought cheaply. Just find a supplier near you, or buy on the internet.

There are also ready made oscillators for this frequency, which is apparently a popular frequency for certain 8051 microprocessors (first link).

Another way, divide a 32kHz watch or clock crystal by 2^32 or some other binary number to get ticks, and use these ticks to run the clock through an interrupt service routine.

Not knowing how this works, it is possible that there are software issues like missed interrupts every so often (say as the minutes roll over) or a change in some constant compared to the original. You didn't mention whether the crystal was correct frequency or not. I expect it has to be divided by hardware (a counter timer) to get it slow enough to count ticks in software using an interrupt routine, say every second or every hunred ms or such. It is the divide ratio of this counter timer that should be changed.

Crystals are manufactured with an error band of 0.1% That would mean that if you have ten crystals of 10 MHz, you might find that when you place any one, the frequency could be lower or higher by 0.1% of 10 Mhz which is 10 kHz. A loss of 12 secs in 3600 secs is 0.33% and I would think is on higher side. But the cost of crystal can be low.

there is a way of setting the crystal to right frequency. Crystals generally will give a lesser frequency and I am sure you will see a loss of 12 sec in 1 hour, not gain. If that be so, the frequency has to be increased. Placing a variable capacitor of 10 to 30;pF in series might help. This can be then used to ensure that the loss or gain is less than 1 sec in 1 hour.

It is also possible in some cases to change the delay number in software to do such calibration . That is if you write the software. A bigger loop (say 1 sec) ADDED to a smaller loop (say 10mS)

will enable adjustment. by adjusting the factor or count in smaller loop..

http://www.best-microcontroller-projects.com/ppm.h...

You could replace the oscillator with a commercial one such as http://parts.digikey.ca/1/1/432181-oscillator-11-0... . Failing that you could look at using a phase locked loop instead of a crystal oscillator. If you choose to use a replacement crystal instead make sure it is a crystal and not a ceramic based device.