Bindweed... grrr

The only organic way is to keep digging, or hoeing it off - assuming it's not practical to cover up the area. 

If that's not possible because it's too close to your veg, then try putting a cane into the soil for it to grow up, then paint some glyphosate-based weedkiller onto the leaves (spraying runs the risk of damaging your veggies) so that it kills it that way.

Pernicious weed, I hate it too, even though it has a lovely flower, it's a lifelong battle to remove it. Glyphosphate is only effective in killing off the root if it is applied generally while the bindweed is flowering, hoeing and digging will eventually discourage the root so that it moves off and comes up somewhere else! Last year I had a lot of it in a spinnach bed so I cut off the bottoms of some plastic Coke/lemonade bottles and put them over the weed shoots, when the weed had reached the top I nuked it with 'Roundup' through the the bottle tops, no overspray and a dead weed.  Had a lot of enquiries about what was growing in my 'quaint greenhouses' and a few sniffs from the green wellies, but it worked for me.

With you on that hightide and Ice. Just bought myself a handspray unit of glyphosphate to spot spray the now appearing bindweed. A close shot of the stuff right on the new young leaves will set it back a lot. When it appears again, another shot and soon I'll have it under control again. Every time I weed or rotovate my garden I spend a lot of time hand pulling the roots but they are so easy to miss, just one bit missed and you have a huge spread. 

I use a parrafin 'flamethrower' - it's an ancient thing that you heat up first (like a primus stove). It is great fun and very satisfying heating up the base of the plants to kill them off Mwaahahahaha!  Have to stay away from the dry stuff tho - don't want to start any fires!