Can a small specialist software site move from long-tail to broader keywords?

[Sera]

I'm looking for some advice from people who have worked with SEO for small software sites.

I have a small website for a very specific type of business software. Google has indexed it and it is already appearing for some very specific searches that accurately describe what the software does.

The problem is that I'd like to gradually reach broader searches in the same area, for example "timekeeping software" and "timesheet software". These are obviously much broader and more competitive than the searches the site currently ranks for.

I've been adding pages covering closely related topics rather than trying to target completely unrelated keywords, but I'm not sure what the best next step is.

Would you concentrate mainly on building more topical depth, improving existing pages, building relevant backlinks, or something else?

I'd be particularly interested in advice from anyone who has taken a small specialist site from very specific long-tail searches into broader competitive terms.