If the company ends up with less cash, then its market cap should be lower, not higher.
When a share is bought back by a company, accountants handle it by either cancelling the stock or turning it into treasury stock, which has negative equity value to cancel out its positive nominal value. In either case, the repurchased stock does not add value to the company.
To illustrate the point, consider creating a company that consists of three $1 bills. You create three shares of the company and sell them to three investors. Assume that the market is efficient and that each share is valued at $1. Let's imagine the company's balance sheet. It has $3 of assets (cash) balanced by $3 of equity (stock). Now the company decides to spend $2 on share buybacks. It spends $2 of its cash pile buying two shares from two investors. Let's look at the balance sheet now. It has $1 of assets left (cash) balanced by $1 of equity (the remaining outstanding share). If you like, you can optionally record the two shares the company bought as treasury stock, so that the equity is $3 of stock and -$2 of treasury stock, for the same total of $1 outstanding stock.
As you can see, the value of the company falls when it executes a share buyback. This makes sense, because after the buyback, the company is poorer.
