Let me get this straight. You are measuring parts that have a 40" diameter. And you are trying to measure that +/-0.001"? Really? For one thing that means that your π tape must be accurate to π x +/-0.001" or +/-0.00314". For a 40"diameter that π tape would need to be at least 126" long - over 10 feet. And in that 10 feet plus, the total error must be less than that 0.00314". Conventional wisdom says that your instrument should be TEN times more accurate than the measurement that you are trying to make. So we are actually at +/- 0.000314".
But lets put that aside for a minute. I just measured the thickness of the tape in a tape measure that I keep in my pocket. It is 0.005" THICK. Is that the same as your π tape? I don't know but I can't imagine that your tape is thinner than that. A 10 foot tape can't be too thin. I will go with that 0.005" number.
You have a 40" diameter part and then you are wrapping the π tape around it. So, if the π tape is 0.005" thick then the OD of the π tape is going to be 40.010". And the markings on that π tape are on that outside surface, not on the inside surface which is against your part.
Now, what are you going to use as the actual diameter that this π tape is measuring? Is it the inside surface diameter of the tape, assumedly 40.000"? Or is it the outside surface of the tape, assumedly 40.010"? Or perhaps, if the π tape has compressed in the inner 0.0025" and stretched in the outer 0.0025", is it the diameter of the mid point of the π tape which is assumedly 40.005"?
This is a range of 0.010" or a full TEN times the accuracy that you say you are getting. But shich of these diameters Is your π tape specified for? Or did the manufacturer actually provide such a specification? I strongly suspect that they DID NOT.
That is only one of the sources of error that the use of a π tape necessarily implies. Yes, the difference in the coefficient of thermal expansion would need to be taken into account. What about the amount of stretch in the tape due to the tension that must be placed on it to hold it against your part. What about the fact that there is another error introduced with whatever method is used to bring the zero mark at the start of the tape into alignment with a mark at the 10 foot plus which indicates the diameter. You must offset the wrap or overlap the two ends or both. Any method you use to bring these two marks on the π tape together to read them. Oh, and I put aside the basic accuracy of the markings on the tape. But that source of error is still there. How were they placed on the π tape? At what temperature? What printing process? Etc.
Frankly, after taking all the possible sources of error into account, I just do not see any way that you can assign a level of accuracy of +/-0.001" to any use of a π tape, much less one at a 40" diameter. I think somewhere around +/-0.010" would be VERY difficult even when all possible sources of error have been minimized.
I think that +/-0.001" on a 40" diameter would be difficult using a micrometer in that size range.