Though I'm guessing (since I don't have this problem, in fact nurses love my veins!

) would being "hot" help with your veins & blood flow? I suspect it would as your body naturally pumps more blood to your arms & legs to cool down and thus also "inflate" veins.
The problem might be being made worse by us going into air conditioned rooms after being outside in hot weather (or being rapped up in a thick coat?), our body would then sense the cooler change in temperature and probably reduce blood flow to all limbs in order to maintain core temperature.
We all know about good hydration, I also believe Omega 3's (eg. fish oil) are useful too as they are natural blood thinners that would therefore help the blood flow, though you need to take them for at least a month before they will be effective, it's no good just popping a couple before your venesection.
I know body builders take nitric oxide supplements as it is a vasodilator, it makes their veins bigger and flows more blood to muscles (though frequent (daily) use reduces it's effectiveness), but that would not be a problem for venesections as you'd just take it infrequently prior. It's easily available at most supplement stores like GNC, though many brands may have a lot of extra ingredients that we don't need so probably go for a basic one. Wether this would help problems with venesections or not I
don't actually know but it may be worth a try if
desperate. As normal, consult your doctor if you have any other health issues.
Also high levels of Vitamin K will increase blood coagulation and possibly reduce flows. Vit K is found in leafy greens, broccoli, cabbage, saturated fats (incl. butter), liver & kidney (bovine only?), there's even some in avo's & kiwifruit. Some live bacteria in some cultured foods like yogurt & some cheese (soft?) can synthesize vit K in your gut.
Note: One study found that high levels of iron (not sure what they mean by that) reduced platelets thus making blood thinner (the same thing fish oil does), though other studies have found the opposite, I believe.