I want to add a new long distance hiking route to the map

Here's what I'd do for a new (to me) walking route in the UK.

Normally I'd add these when I find a route signpost that I've not seen before. In your case you know it's new as you're creating it :) but I'd first check the relevant section of this page (other countries I'm sure will have similar pages). If it doesn't exist there I'd check other sources where possible to see the route, and check OSM in those areas to see if anyone has added anything (searching is tricky since people may use names that vary slightly).

If they haven't, the next question is - should it be in OSM? I'd argue that temporary and unsigned routes shouldn't be included, but permanently waymarked ones should. This may well vary by locality, though.

If it needs to go in, create a new relation by adding part of the route to a new relation (how to do that will vary by editor). The tags on the new relation should be similar to on this one for a new foot/hiking relation (personally I'd use route=foot on shorter, less arduous ones and route=hiking on others; other people may have other ideas). Don't add the route name to any way names on the route since the ways themselves aren't called "route name", only the route is.

Depending on the length and status, you'll probably want to add "network=iwn", "network=nwn", "network=rwn", or "network=lwn" as appropriate. There's guidance on that (and on other tags) here.

Then I'd add a row to the appropriate table here, with internal and external links as appropriate (I can never remember how to do these so always end up copying from another row). Your new relation number goes in the 5th column in the "browse relation" thing. When you've filled in the wiki captchas at save, check that the page formatting is OK and the links work. If it makes sense to, I'd also create a wiki page like this one (I created that because I wanted somewhere to show a picture of the sgnpost).

Once you've added all the ways to your route, you can see that the result shows up here, and you can use the "analyze" link on the wiki row to check that you haven't left any gaps.