Not sure I understand your questions. If an extension option isn’t included in the original lease and the lease is extended through an amendment, or in the form of a new lease, an amendment or memo should at some point get recorded in the Reeves County deed records. Like original leases what is recorded typically won’t show what was paid for the extension.
Here’s a link where you can search the Reeves County Deed Records. https://reeves.tx.publicsearch.us/
I’m not sure how that fits with your other question about whether you can lease to another company. If your current lease didn’t include an extension option and the company holding your lease is trying to negotiate an extension, you shouldn’t have to agree to their request, and in that case your lease would terminate at the end of the primary lease term. You would then be able to sign a new lease provided there is another Lessee ready to do that. But there are things to consider before deciding whether to turning down the offer to extend.
Does the company holding your lease control the other acreage around you to the point another operator won’t be interested in that area (which may depend how large a mineral interest you own)?
You don’t have to accept the current operator’s first offer. Can you negotiate a higher bonus for it, a shorter time period for the extension, or require improvement of other provisions in your original lease that would make it more appealing (which again may depend on the size of your mineral interest)?
Bottom line, don’t decline the current request unless you know it is the best you can work out with the current operator, or you have a better alternative.