On solution may be ( but this not guarantee you any sale ), to find old websites, re-design them and after that send an offer to the owner(s). For example, in the lawyers niche you can find a lot of old design websites. How do you find them? Go to attorney web directories, such hg [dot] org, and search them. Make a list with the old websites you find and start the work.
Before you will start the effective work, search for this:
1. The owner is still in the business - To find this, take a look on it's social media accounts ( Facebook, Linkedin, Twitter, etc ). If it's still active there, there is a chance to get positive results.
2. The owner doesn't have another new website, already re-designed - In some cases, the owner simple let the old website and get to a new one ( with a new domain name, new design, etc ). Make sure that what you've found is not already redesigned. How do you find this? Simple: Paste the owner contact details ( phone number or the office name + location ) and see what Google reveal. If there are two websites with the same info, the chance to get it as a client become very low.
After the two steps above, start to redesign the website:
1. Put it under a subdomain of your website. Make sure that Google can't crawl it. In this way, you can avoid some copyright problems. You only want to show it to the owner. You can also put it under a "password protected" feature.
2. Redesign it and use the info from the old website - Because you are not 100% sure that the owner will buy the new website, don't work too much at the new design. Include only the most important content. This may be: Homepage, About ( or Our Attorneys, etc ) and Contact page ( most of them have static contact details, without any contact, map, etc. Try to improve this page ).
Contact the website owner, with your best quote. Make sure to include the full offer and some other freebies ( free hosting, free seo, etc ). If he don't want to buy the website, don't delete the website. Keep it there, password protected, in the idea that he will return. You don't have anything to lose.
This idea don't guarantee you any sales, but it's still better than nothing. If the new client is satisfied, he may recommend you to other people.
Cheers