I agree - 60sec per day is 700ppm, that's too much for adjustment.
But clockace's 5sec per day are possible. If the trimmer is at its end, you can add more capacitance to the oscillator circuit. An additional 10pF capacitor one each side of the QC should suffice. There should be 2 capacitors already on the board, so you can either replace these with larger ones or solder 10pF in parallel of each. No guarantee that this works, maybe the oscillator is already at its limits, but it's worth a try.
Each QC has a "load capacitance" specified, typically 12.5pF for 32.768kHz QCs.
If you replace the QC in a circuit that was desinged for a QC with different load capacitance, it will oscillate (slightly) off frequency. The load capacitance is made of two capacitors, the trimmer being one part of it. A good description is
here (PDF) or
here (Wikipedia).