Collapsible trekking poles

I know you mentioned that you don't have room in your bag and don't want to check it, but there is another option. Check only the poles. Put them in a small duffel or box wrapped in cardboard or bubble wrap. Ditch the packing material at your arrival airport. You might find more cheap options in the U.S. and if you happen to be flying an airline that doesn't charge extra for a checked bag, this could be cheaper and save you the time of looking for a shop.

As a point of reference, I bought a collapsible hiking pole in France on my last trip and it was 28 EUR. It folded like a tent pole rather than collapsing like a telescope so that made it quite easy to carry in my bag (although I did check my bag). It was similar in style to this:
https://www.amazon.com/TOMSHOO-Aluminum-Adjustable-Telescopic-Trekking/dp/B01FHXVG70/ref=sr_1_19?s=sporting-goods&ie=UTF8&qid=1501637028&sr=1-19&keywords=folding+hiking+pole

On a few occasions we have simply purchased inexpensive trekking sticks, used them for our trip, then "donated" them at our last stop. Usually paid 20 to 30 Euro pair. We have now purchased truly collapsible poles that both fold and telescope so I can fit them in my daybag, but if you are not going to need them over and over and do not wish to carry them, just buy a Lois end set and make a donation.

If you are not going to pack it, why does it have to be collapsible?

Ask your accommodation if you can borrow a trekking pole or an ordinary walking stick - maybe someone has donated one earlier.

...why does it have to be collapsible?

An added advantage of "collapsible" trekking poles is adjusting the length to fit the terrain. On steep climbs I like to shorten my poles compared to striding on flat terrain. And of course, if you need to scramble, collapse the poles and strap them to your rucksack.

https://www.rei.com/learn/expert-advice/trekking-poles-hiking-staffs.html

For general hiking, adjust the length so that when you hold the pole
with the tip on the ground near your foot, your arm makes a 90-degree
bend at the elbow. This will be the right length for most of your
hiking.

For long uphill sections, you can shorten each pole by about 5–10cm to
get more leverage and more secure pole plants. The steeper the slope,
the more you shorten your poles.

For long downhill sections, try lengthening each pole by about 5–10cm
from the length you set it at for general hiking.

If you’re on a long traversing section, you can shorten the pole on
the uphill side and lengthen the pole on the downhill side as needed
to improve comfort and stability.

Thanks for all your suggestions.

For Cinque Terre, in the town of Manarola, is a small outdoor shop halfway up the main street called Cinque Terre Trekking. I believe it's owned by a Italian husband and his American wife, they put on a number of local races and a great spot to get info on the local trails or, to get socks/shoes/poles/hydration or a souvenir t-shirt.

For hike poles, I'm a fan of Black Diamond's, three stage collapsible, easy to pack-up while sturdy and comfortable.

Thank you zcorsair! Will check out the store in Manarola.

Just checking back...there's also stores for both The North Face and Patagonia in Milan; being company stores they may not have trekking poles but, they should be able to direct you. I would imagine the closer you got up to the Lakes District, there will be outdoor shops available also.

A bit off-topic, when I was in Bologna, I ran into at least three outdoor shops around the Centro Storico.

Thanks again! I'll check the stores in Milan.