Will more hashes per second increase bitcoins per hash per second?
The short answer is: No. The average number of bitcoins per hash is fixed, and is roughly equal to B / (2^32 * D), where B is the bitcoins per block and D is the difficulty. Every calculated hash is either a block or not; if it's not, it doesn't bring you any closer to finding a block.

However, for completeness it should be noted that there are several sophisticated attacks that allow superlinear gains on hashrate, such as lie-in-wait and hashrate amplification (also known as selfish mining). Superlinearity can also be a side effect of a short block interval. All of these require huge hashrate to be effective, on the order of a quarter of the network hashrate - not something which should be of concern to the average miner.