Does your borrowed scope have an illuminated reticle? If not, then this may be part of the problem. Using a BAC ACOG (like the TA11E) at close range, you keep both eyes open. Instead of trying to look through the scope, just keep the scope where the reticle is visible but scan for the target. When you see the reticle over the target, fire.
You are basically using binocular vision to trick your brain into combining the wider field of view from the non-dominant eye with the bright, high-contrast BAC reticle from the dominant eye.
Read the general purpose optic thread (if you haven't already) as there is a wealth of information there, especially about BAC ACOGs. The post concerning optical health issues that might make an ACOG less desirable is worth a read before dropping the cash on an ACOG.