6-9 volts is in the range I wouldn't suspect a lost neutral problem but is in the range of maybe a voltage drop problem of the feeding neutral to this barn, or the service neutral, and even could be the voltage drop of the utility's MGN.
If this is a typical farm type service where you have the meter and disconnects all located on a pole with feeders to each building then most likely you only have 3-wire feeders to each building, if turning off the power at this pole for the affected barn eliminates the voltage this will indicate that the voltage drop is on this neutral, if this doesn't eliminate the voltage then it is the voltage drop of the service neutral or MGN.
If you determine it is the neutral feeding this building, you have but two choices.
Choice 1: to me would be the best choice is two add a separate EGC from the service disconnect to the barn, separate the grounding and grounded conductors at the barn as would be done for a sub panel, this will keep the neutral voltage drop from appearing on the grounding for this barn.
Choice 2: calculate the correct size neutral conductor as per load to lower the voltage drop, not the best thing to try to do since farm animals can sense voltage as low as .5 volts.
If you determine that the voltage drop is ahead of the service point, you might have to get the utility involved to isolate it to see if it is the service drop or the MGN, this would be done by removing all power from the service then checking to see if the voltage is still there, if it is then its the MGN, if not its the service neutral.
How far is the serving transformer from the service disconnect?
If it is over 200' you might be able to get the utility to add an isolated EGC from the transformer to the service, but because most meters have the neutral bonded to the can, it is very hard to find one that can be isolated, again separating grounding and neutrals at the service as if it was a sub panel would be a must.
Relocating the transformer closer to the service point is also an option but can be hard to get done if there is no closer right of way for the primaries.
If it turns out to be that the voltage is in fact coming from the MGN, the only method I know of ( that Will be cost effective for the serving utility) is to have the utility to install a neutral block device at the transformer, this isolates the secondary neutral from the primary neutral (MGN)
I highly doubt that the voltage is from the HV lines passing above this area, for the most part this capacitive coupled voltage will be shunted via the earth in short order, as for voltage from the tower legs, I cant see how any voltage can be on these legs as these towers have multiple points of bonding to the static conductors that run with the HV phases, and each tower is bonded multiple times to very long and deep grounding electrodes installed at each and every tower.
Will voltage build up on a conductor in close proximity to a HV transmission line? sure it will if it is isolated, but shunt it to earth even with a very high resistance load and it will go back to zero.
Also the fact that voltage does not travel very far through earth before being brought to the Earth potential of 0 volts.
It is the problem of other conductors that are used in the grounding path such as the system neutral that have a voltage drop on them when current is applied that will raise the voltage to a point above Earth that will give you this difference of potential.