Indeed support period very important. A Website is not a “One-Time-Shot”. Professional websites have to be monitored, updated and extended on a regular basis.
Choosing the best software is important eg., The TYPO3 CMS gives good LTS support (4/5 years) for a particular version.
We had such also phase such issues in past with our customer because they were not serious about it at the starting of project. We have included this in our standard proposal eg., Small site website = 1month, Medium to large = 2/3 months (based on site). Also, at the initial stage, we offer the Support & Maintenance proposal too. We are trying to learn & aware about such important factor of "consider support periods" by choosing long-term horse software (like TYPO3).