Which methods aside from user testing are suitable when designing for product re-design? Participatory design is an option but wanted to explore some other methods.
One of the best and easy ways is to go through all the reviews of products in that category.
for ex. If I had to re-design a furniture website, I would go through google reviews of all furniture websites and stores, this gives a lot of genuine insights about a user and can help you come up with amazing insights, features and ideas for re-design
To add to the above answers, It is good to do analysis on what are the pros and cons of an existing site, moreover, you can do SWOT(Strengths and weaknesses, opportunities and threats) type of analysis. Identifying your website's strengths and weaknesses will benefit on new designs.
What do you mean by optimization?
I don't think participatory design is the right way to design for optimization . Participatory design is a way to get buy-in / permission to make design changes from other people within the company.
Optimization usually means you are trying to improve something when compared to different options, it's a quantitative method. It's usually done via A / B tests with few variables as possible. You do solution A x B, check that B is successful, then you do B x C check that B is successful, then you do B vs D and check that D is successful.
IF you want to improve a product in general for the user you could use a north star metric. Example, for something like Uber it could be completed rides per user. The product is usually "successful" when it becomes sticky (habit-forming).