While it is useful from a negotiation standpoint to know that there is money on the table, it does little to convince the employer to give it to you.
First of all, to a certain extent, the OP does deserve to be at the low end of the salary range for a senior manager. He just started the job! Should he get paid as much as a senior manager who has been a top performer for several years?
If the OP seriously believes that he is underpaid based on his position, he needs to be able to articulate why he believes that. Is his salary significantly lower than what is indicated for his job in online sites like Salary.com, Glassdoor.com or bls.gov? Is it significantly lower than what one would expect for his education level and experience? Armed with that, then you can then approach whoever at your company manages your career.
The “I’m not going to quit but I want more money” negotiation strategy almost never works. But what you might want to try is to discuss with your boss why you feel your salary is below market (based on above) and then come up with an agreement for some specific milestones or acomplishments you can do to get your salary in line with what you think is fair.
And if they don’t follow through on their end, you have your answer.