Honestly, I must admit that I'm not an expert with Surly as they're very rare in Europe. I'll assume it's Shimano equipped.
However, based on your question, I would start with the simplest option and make sure that you have the widest cassette possible before changing anything else. This will depend on the capacity of your rear mech (how much can it handle).
You'd want the largest spread possible ... 10-51 is quite awesome.
I would wager that your drivetrain can't handle a 12-speed cassette, so ignore that first option, but that 11-46 range is pretty good.
https://www.bike24.de/1.php?content=...31%5D=1;page=1
The guys here could probably confirm what you currently have and if this idea would work.
If I believe the Surly website, you have a
Shimano CS-HG50-10, 11-32, 10-speed
And would see a huge improvement with a 40, 42, or 46-toothed cassette.
I believe (someone else confirm) that this would be directly swappable and should take a bike mech about 15mins to do it and readjust the rear mech.
https://www.bike24.de/1.php?content=...31%5D=1;page=1
Just call up a local shop and tell them you want a
Shimano CS-HG500-10 Cassette 10-speed - 11-42 Teeth andask for a quote.
Cheap and easy for about €25 for parts and maybe depending on US labour rates at a shop, out the door for under $100.
I just think that's an easier option than adding a triple to the front (you may need new shifters and it's a huge amount of labour compared to a cassette swap). Disregard if you have barend shifting.
Reviews on this extended range cassette:
https://www.merlincycles.com/shimano...tml?source=PHG
Good luck and keep on riding.