Do you already know where you'll be staying? I assume you will still have a car to get to Montalcino and move around, to see the area in any case since many places are not in town.
You are right that the majority of tours start in Florence or Siena, but with a bit of patience, you can most definitely find tours that start in Montalcino. Here are a few I found for you, with brief descriptions of what they are (disclaimer: they are affiliate links to tours on Viator)
They are a mix between group tours, generally small, and fully private, you'll see by the cost....
Wine Tasting group tour to 2 local wineries - 3 hr total, morning or afternoon - not a private tour, but small group, 8 person maximum
Brunello Wine Tour - 4 wineries, plus 4 course lunch
Montalcino Private City Tour - 2 hour - walking tour of just the town of Montalcino
Full day Private tour from Montalcino to Pienza to Montepulciano -
Another tour of Montalcino, Pienza and Montepulciano - with wine and pecorino cheese tastings
Gourmet Wine Tour - 3 estates, including 4 course lunch
A different Gourmet Wine Tour - 3 wineries, plus lunch
If you're feeling active and want to do some hiking, which I've done in the area and highly recommend for the incredible views and walk along the vineyards, you could consider the 2 walks or even the bike tour:
Hike Tour about 6km around Montalcino, with Wine Tasting - and option of Light or Complete Lunch
Hiking 8km - you decide which 3 wineries to stop at for tastings (cost of tastings is extra)
E-bike Tour - includes the Abbey of Sant'Antimo, winery visit and tastings and light lunch at the end of the tour
Wine Tastings and Visits - 3 wineries + lunch - picks you up to visit the area, but tour departs from Siena - small group, not private unless you wish it to remain just your group
Cooking Class + Dinner + Wine tastings at local winery - learn how to make pasta, get the winery tour, then sit down for 4 course dinner and tastings with each course
These last two require you get there on your own, of a local winery and a cheese producing farm:
Wine Tasting and Lunch at Local Winery - about 20 min walk from the Fortress in Montalcino, right outside the town -
Visit and Pecorino Cheese tasting at local farm nearby - this requires a car to get there, it is between Montalcino and Buonconvento
Have fun! I am sure you'll love the romantic honeymoon in Italy!
