Hi,
Although I've helped many SaaS startups, your question is a bit too general to give an educated answer, because it really depends on the type of service you are providing, the niche' you're targeting, the clientele, country, etc. For example, launching a SaaS product for B2B isn't the same as launching for a B2C product. A product for younger consumers wouldn't b launched the same as a product for older ones.
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There are multiple approaches for a SaaS product launch. The main problem you will find is finding your first few clients. This is due to the fact that most potential clients look at your current portfolio and would want to see if it is worth their time using yours. If you have no clients, this process can be quite long and de-motivating. Your best bet is to give out your software for free at first to select clients, to build up a portfolio of active clients that you maintain. From there on, you can go out and sell it to bigger clients.

Unless you have established track-record and connections, launching a new SaaS product, can be quite difficult. Therefore, it is important to get good referrals as quick as possible, by giving your first clients a free try.

Before you finalize the SaaS product launch, make sure that the launch is scheduled around a specific date like a customer event, an anniversary etc but do not develop it in vacuum, discuss it with product, marketing, sales, customer service, and executive teams. Your customer must be forefront of your mind because he will give you feedbacks to innovate yourself if the product is not functional. Therefore, it is incredibly important to beta test with current clients.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath