The Verdict: Unlikely
Most of Ireland's working holiday agreements allow one authorisation per person, so a second one is unlikely. Check your own country's rule on the embassy page before applying, and look at an employment permit for a longer stay.
Australia, New Zealand and Argentina refuse anyone already granted one, even if it was never used. Hong Kong allows a new application only if the earlier one was refused. Canada has two categories, the working holiday and the co-op internship, and you can do each once. The United States allows another once your previous authorisation has been expired for a year.
For Chile, Japan, South Korea and Taiwan the embassies do not publish a rule, so ask before you apply.
An employment permit is the usual way to stay longer, starting with an Irish employer's job offer. A parent or grandparent born on the island of Ireland would let you live and work in Ireland without either.