Based on the last 3 years your forecasts are way too low for the peak months ... thus you are right to look for a better method/approach.
R Time Series Forecasting: Questions regarding my output presents a similar question. Essentially daily data ( particularly when there is a human element) is often driven by a combination of memory and events. Events are not only holidays BUT day-of-the-week , month-of-the-year, day-of-the-month etc.
ARIMA models easily incorporate empirically identified pulses ,level shifts and local time trends while incorporating parameter and error variance changes. Additionally ARIMA models easily incorporate user-specified causals like weather morphing into Transfer Function Models or SARIMAX models . How to predict the next number in a series while having additional series of data that might affect it?
You might want to look at this reference http://www.autobox.com/cms/index.php/blog/entry/advantages-and-disadvantages-of-using-monthly-weekly-and-daily-data to more fully understand why you need to be using daily data
If you post your data and tell me the kind of data that it is and the beginning date I will try and help further.
Pursue this thread https://stats.stackexchange.com/search?q=user%3A3382+daily+data for more readings and examples of how to model daily data.
EDITED AFTER RECEIPT OF 1559 DAILY VALUES:
The Actual/Fit and Forecast for AUTOBOX is here with forecasts here
The model that was automatically developed was rich in structure.... holiday events were significant , month-of-the-year significant, day-of-the=week significant and two level shifts .
ALSO presented here and here
The residuals from the model suggest randomness
Another view of the model is here and here and here
In summary :
Temperature is very significant
Holiday effects ( lead and lag around holidays ) are very significant
Month-of-the-year is very significant
Day-of-the-week is very significant
Two level shifts are significant
Forecasts were developed for daily temperatures which play a significant role suggesting that one might prepare more accurate forecasts for temperature using different environmental scenarios. This might explain the "low forecasts" going forward.