The answer is probably, yes. As a business owner thinking about marketing, you want to consider any available channels of communication with your customers, and communicate with them through the methods that they prefer to use. If your customers are on Facebook and using it regularly, then the chances are they would be receptive to hearing your news there too.
Look at the range of social media platforms that are available. Facebook primarily attracts personal content, whilst LinkedIn is effective for B2B networks. Pinterest and Instagram are the go-to platforms for visual content, and Twitter is good for sharing thought snippets and status updates. Any of these can be used to direct attention to your new website content, and whilst you might want to focus on the one that resonates best with your audience, you should probably have a presence on them all.
But don’t think of it as another cost. Social media activity, and your whole online marketing effort should be focused towards providing content and answers to questions that your customers, and potential customers, are looking for and want to see. The joy of digital marketing is that it is all quite low cost, other than the time you invest into it. If you regularly share valuable, insightful content, then you build authority in your field which reflects positively on your search engine performance, and ultimately your business.