It’s neither:
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What I’m looking into would be a small set of professional services offered with a fixed price, scope and “feature set”.
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The deliverables would be PDFs, e-mails, Skype/phone sessions, etc. There may be software involved in delivering the products or the processes in which the deliverables are created, but the product itself is not software.
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Most services (the ones I plan to start with) would be one-time purchases, but I am also looking into a recurring/subscription-based service.
It could be doable, as I would probably not have to leave work for most issues. I’m already sitting in front of a computer all day anyway, but I would of course not be able to bill the “day job” client for any time spent on my own products. However, potential time/resource conflicts with current client work is one of the reasons why I am not looking into launching a SaaS at this time (and also the fact that I don’t have any useful product ideas for a SaaS, yet).
Being available for short incoming phone calls during my regular work hours is not a problem. I would typically not be replying to e-mail, but I could if something required a quick reply. Scheduling a one hour Skype meeting during the day would also not be a problem, as long as it’s planned a few days ahead. My “day job” work schedule is very flexible, but any “irregularities” would have to be clarified with the client, as they need to know when I’m available if I’m not at the office or if I have to step out for an hour or two. What I cannot do, however, is spend hours each day doing lead follow-up and outgoing sales calls during office hours.
I also have the option of working at my home office one or two days a week, so I could set up one or two days with regular office hours for “product” customers, if need be. However, most of that time would have to be spent working for my main client, but whether those hours are spent during the day or during the evening, they do not really care as long as I’m available to them for questions/clarifications during the day, or if a production incident occurs.
I don’t know what service level an UpWork freelancer selling “off hours” is offering, but I am not planning to target customers who want the cheapest solution. Customers who want to spend the time and effort, and have the technical skills to do so, to try to save some money by going the UpWork route for the services I plan to offer are not my in my target market. I do however, plan to target customers who have gone this route and maybe experienced that the end result was not of the expected quality, and customers who do not have the technical skillset to do this themselves.
To summarize, the service level I can (easily) offer, even while working a full-time consulting gig is:
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Replying to any e-mail request within 24 hours (in practice, usually faster).
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Being available via e-mail, Skype and phone between 09:00 - 21:00 CET seven days a week for short calls, chats or urgent e-mail replies.
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Being available for longer Skype/phone meetings between 09:00 - 23:00 CET seven days a week, if they are scheduled 1-3 days in advance.
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Being available for in-person meetings/workshops any time on weekdays or weekends, if they are scheduled 3-5 days in advance for customers in Europe or 1-2 weeks in advance for customers outside Europe.
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Delivering product deliverables within 3-7 business days (depending on service package purchased, and size/complexity of project).
That being said, the services I plan to offer will typically not be (operationally) mission-critical or time-sensitive.
Also, thanks to @Ken for excellent background info on productized services. 
