Almost all the time in a jigsaw goes on searching, so everything that works is a way to shrink the pile you search.
Most advice about solving jigsaws faster is really advice about sorting. That is not wrong — sorting is most of it — but it is worth understanding why, because the reason tells you when to stop.
Placing a piece is quick. Finding it is not. Almost all the time in a jigsaw goes on searching, so every technique that works is a technique that shrinks the pile you have to search. Sorting works because it turns one search through a thousand pieces into a search through forty.
Which also tells you when a technique stops helping: the moment the sorting costs more than the searching it saves.
The border is not easier because it is the border. It is easier because a flat edge is a property you can see instantly, on any piece, without comparing it to anything. That makes it the cheapest possible sort — and it gives you a frame that turns "where does this go" into "where does this go within a known rectangle".
Grouping by "sky", "roof", "grass" feels natural and is slower than it should be, because it makes you interpret every piece. Grouping by dominant colour is almost mechanical, and it gets you to the same place. Interpret later, when the group is small.
A face, a sign, a bright object, a hard line between two regions. These are the parts where a piece can only go in one place, so they build fast and each placed piece constrains its neighbours. Growing an existing island is always faster than starting a new one.
Sky, water, snow, plain walls. Every piece you place elsewhere shrinks the number of gaps a sky piece could occupy, so the same piece that was impossible at the start becomes trivial at the end. Attacking the sky early is the single most common way to make a puzzle take twice as long.
In a uniform region, colour tells you nothing and shape is all you have. Look at the gap rather than the piece: count the tabs and blanks it needs, then find that. It is a much smaller search than the other way round.
The puzzle of the day is identical for everybody, so it is the only fair way to compare a time — against other people, or against yourself last week. Same picture, same cut, same count, and the clock runs on the server rather than in your browser.