Some people have successfully cut back end of an x1 slot with a dremel tool or soldering iron to fit an x16 card in it and accomplished what you are trying to do. I'm guessing the adapter would prevent a full size card from fitting? Can't say I've ever seen an x1 to x16 adapter.
I don't think that a 1x slot will give you the bandwidth you need for a video card even though it is not the primary card. I don't know about a x1 to x16 adaptor and what it looks like or how it fits in or over a x1 slot. Your Nvidia cards will not support triple monitors with a x1 slot involved. I'm guessing a new motherboard is not in your budget.
There are x1 to x16 adapters for servers, but not likely to be helpful in this case. Also, the are x1 video cards (typically for workstation class GPUs). To support more than 2 monitors with a single GPU requires an AMD/ATI GPU.
That thread might help you if you are willing to physically modify your slot. 1x is plenty of bandwidth for a secondary card that's just running a 3rd display as an extended desktop.
That being the case why the hell do they even make x1 slots if the bandwidth is suficient , why not make all x16 slots and just determine which are x16 , x4 and x1. Actually some MB are coming out like that.
Well if you look at his board you couldn't fit x16 slots where those x1 slots are, there are some caps and a bios battery in the way. Not sure why they make most x1 slots close ended though. Lots of boards have 16x slots that are only really 4x but I can't remember seeing x16 slots that are only really x1. I guess there is still enough of a market for x1 devices like sound cards that board makers decide to include them.
If the monitor connection on the full size video card is on the lower part of the card just remove the back plate of the video card and cram it in. Use a zip tie to hold it together. The upper half of the card will not line up with the case so you will not be able to screw it in or use the upper port if its there. Chances are the case cover wont fit.
Then again that adapter looks to be only half an inch so it might not be too bad.
If your card is full size I don't think that adapter is going to work for you, unless you can modify your case somehow.
+1 to benski's observation here. There are adapters that can be used (made for 1U servers) to turn the card sideways. This will create a different dimensional challenge.
While I applaud's the OP's efforts to use the tech available, this is quickly becoming a project that will result in more hassle than benefit. Just an observation, and not a criticism, but there are easier (and cheaper) ways to skin the multi-monitor cat here.