How low of the cartridge output the Allnic H3000 can handle?

As a stopgap, I’m using an old Monster Alpha Genesis for the moment- an ancestor of the ZYX--at .2mv output which is pretty low. No issue on my H3000- I did adjust the loading so that it is a little over 100 ohms, compared to my usual Airtights, which load down closer to 70 ohms. A little less gain at that setting- by 2 db I think, but not a major issue. I’m planning on installing a Koetsu next. Should be no issue.
PS: within the range of settings, it is pretty easy to experiment on the Allnic with the twirl of a couple of knobs - mute your phono output temporarily when you change settings. You can easily select the best loading pretty fast, and keep experimenting as time goes on without much trouble.
PPS: did you roll the power supply rectifier on yours, Calvin? That seems to make a big difference in the performance of this unit and there are plenty of anecdotal reports on the web about various rectifier tubes that work. Most aren’t terribly expensive even for "NOS" compared to the price of vintage small tubes these days.
BTW, the importer in the States (if that's where you are) is the lovely Albert Porter, who is a gem and has lots of hands-on knowledge. 

Calvin- I've never used a Hytron- some look like Sylvania post WWII, others appear to be like Tung-Sol. I'll have to read up. In the meantime, here's a fun piece about the rectifier compiled by someone known as DubStep Girl who listened to a lot of them. I've had several. Right now, using a GEC U52. Her impressions matched mine to the extent I've tried some of the ones she mentioned. 

There is a WWII era military RCA, I think that a lot of folks like and it isn’t crazy money. I’ll see if I can find it for you and post here. I have used old Mullard fat base 1st series, which was OK, not as good as metal base Mullard which is now very expensive too. I do think that Dubstep Girl write up is pretty informative, not that i’d go throw hundreds of dollars at an unknown tube, but at least you have an interesting list of things to consider. More later....

OK, it is a WW2 Double Bottom getter RCA JAN CRC 5R4GY. If you look up Allnic, rectifiers and Rockitman (who does post here occasionally), he thinks these are the next best alternative to the WE, which are tres expensive. The RCA is not stupid money and is probably worth a try. I gather that the WWII era ones are better than the later ones, but take a look at some of the anecdotals. It may also be covered in the Dubstep list too. 
good luck and good listening- enjoy your phono stage! 

Yep, @wrm57  have done the same things- I think we both had Steelheads, no? Nice to see you, wrm!