Try Woodland Hills Wine. Not sure if it has been affected by the pandemic, but they usually ship a whole lot quicker than K&L – usually overnight for local deliveries – and I like their selection better than K&L.
Also try John & Pete’s on La Cienega. They have some interesting stuff and they have free local delivery - same day – although I haven’t tried in the pandemic.
If you’re close to K&L, you’re probably not too far away from Domaine LA. You can order through Postmates.
Had another good experience (with a caveat) using Mercato. I don’t really need anything, but am starting to run low on TP, so searched Mercato and found some at one store, Los Toros Meat Market, of all places. An off-brand–labeled in Spanish–but hey, why not. So I ordered a couple of mega-jumbo packs plus a few food items, got a delivery slot for the next day, got everything I ordered (a bit late but that’s okay), including the toilet paper. HOORAY!
It was a pyrrhic victory, though. The goddamned stuff is scented! 
I found an annoying workaround to that problem — I ordered delivery but did not choose a delivery date because they were way out, as you experienced. But once the wine status changed to “picked” a couple hours later, it did offer next day delivery dates. The system appears to assume the maximum possible time for wine to be ready for delivery, even if you’re only choosing things that are in stock in Hollywood. They should fix but in the meantime that workaround might help. Of course, it won’t if the delivery slots are actually full.
I did get my picked notice a couple days later, but I was too busy working @ home to log in and see if I could get an earlier delivery. As the week drew to a close, I needed some food fill-in so I decided to just pick up some wine @ TJs while I was there and wait until Tuesday for the K&L truck.
I wish Domaine LA was doing curbside pickup instead.
I thought K&L would be fast since they do their own deliveries vs shipping. But I guess lots of folks must be ordering delivery beverages right now.
If you choose items in stock in Hollywood it should go to picked in a couple hours, giving you in effect next day delivery on your purchase. I’m not saying it is a good system — just a workaround that I am living with because I prefer K&L’s selection, service and (in normal times) IT to other local shops. (I live within walking distance of Domaine and am put off by the service there every time.)
Unless you just want wine, any wine, I’ve never been blown away by Domaine’s offerings. I would definitely head to K&L before heading there and even more likely to head to the Internet to Woodland Hills Wine knowing I can have next day delivery and not have to trek across town to Hollywood.
If you “just want wine, any wine,” Total Wine & More is the place. Domaine LA specializes in “natural” wine, which probably looks pretty random if you’re not into that.
I actually just ordered from Total Wine & More as I wanted to send somebody Champagne for their birthday and they could do same-day delivery.
I actually do like natural wines (although not all - some are just funk and junk), put still never blown away by Domaine’s selection. Maybe they are just too small and don’t have enough capital to carry a truly first-class collection.
Natural wine shops all have the problem of limited availability.
What makes a shop “first-class” by your standards?
A large selection of grower champagne (I love you Crush in NYC) and a big selection from my favorite importers like Louis Dressner.
Domaine does consistently have a nice inventory of Cedric Bouchard. 
If Wine House has what you’re looking for, I believe they are doing curbside (and will actually ONLY do curbside or shipping).
Domaine LA doesn’t sell a lot of Louis-Dressner? That would be pretty odd for a natural wine specialist.
Thanks everybody for future wine buying options! Luckily between my TJs pickup and my K&L delivery I should be set for the next several weeks.
Also, just to add to the wine curbside pickup/delivery discussion for those in Silverlake or adjacent areas: both Silverlake wine and Vinovore are delivering in the immediate vicinity and offer curbside pickup. LOU is doing curbside pickup.
@happycat, re the scented toilet paper you managed to obtain (and I had never heard of scented toilet paper - learn something new everyday). On the bright side, a fair number of people who develop COVID-19 (including some I personally know) lose their sense of smell and taste. So if you can still smell the toilet paper (or the roses) hopefully you are good.
Thanks for giving me a silver lining!! 
Meet… FIORA!! My mom loves this stuff…