Have you noticed that when you do a search in a search engine for your library, you often get a little box with a map and directions? This is a business listing for your library. You can claim that listing and add additional information and photos.
You can add hours, email address, website URL, Facebook URL, Twitter, etc. You might also add any information that you think would be of interest to visitors or newcomers, e.g., free wifi, computer classes, story time hours, etc.
In Google, if your library hasn't been claimed (or even if it has by someone in past years but the information is outdated), you'll find a small link marked Feedback. Click on that and it gives you an opportunity to claim the library and update the information. You'll need to verify that you are a legitimate updater by getting and entering a verification code to one of the contact methods listed. So, make sure you're near the phone if that's what you choose. And hopefully, that phone number is correct. One of the challenges for many of our rural libraries is that your mail comes to a PO Box. The mailing verification option wants to use your street address. Oh well, it is the street address that you want to come up in the listing because that's how people will find you.
In Bing, you'll find an option under the search result to Report a Problem. Click on that and you'll see a box where you can Claim this listing. That will allow you to make changes in a similar fashion to Google.
I don't know of many people who use Yahoo! as their primary search engine but it also offers business listings that you can update. It may be worth looking into any other search engines that you're familiar with and seeing what they offer as well. But, remember, you're going to have to keep these up-to-date if they're to be useful so you'll probably want to just choose a couple that you think are most likely to be used by your current and potential customers.
I think this is a good option to supplement your website not replace it. You'll probably want to keep a website where you can offer links to services like your library catalog, Discover It!, current events, etc. But having an enhanced search listing can really be helpful for people locating your library on a mobile device. Besides, it's easy and free, if you can resist the offers to upgrade ;)
So, go to it and share your experiences in comments.