there are different types of resistance wire. they all have some common things such that resistance is not linear and hence it is not easy to measure. it will also depend on temperature etc.
common practice for measuring low resistance value is to use 'big' current.
while the 'big' current doesn't have to be in order of 1000A, anything higher than your meter will be more precise. if you ever wondered, current used by your multimeter is probably much less than 1mA even for low resistance circuits.
try this: use LM317 for example (or 7805 if you want to) to make
current source of say 1A or at least 100mA (this can easily be measured with your meter).
then let this current flow through unknown resistor of low value.
use meter to measure voltage drop accross resistor. use calculator...
R=V/I
if the current was set to 1A, voltage drop reading will represent resistance of the wire. For example 12mV would mean 12mOhm or 0.012 Ohm.
you can use any number (even something odd like 247.9mA) but the decimal values such as 10mA, 100mA or 1A make it easy to read without need for calculator. Note that you should check your regulator rating.
Don't think you can get 10A out of 317L.
the L version of regulator is good only for 100mA so the resistance between Adj and Vout should not be less than 12 Ohm.
Say you use 22 Ohm. You shoudl be getting current of about 54.5mA so if you read voltage drop on that piece of wire (say it's 13.9mV) when this current is flowing through it, put those numbers in the formula above and you will get something like 0.25Ohm...
Now you can go to Fluke and complain