No, the Hyde Amendment does not conflict with women's reproductive rights, because babies have a right to live, too. Some people believe that babies are alive in the womb, and they would feel like a murder using public funds on abortions. These people should not be forced to fund abortions for other people. The Hyde Amendment is an appropriate balance of these issues.
The Hyde Amendment in no way conflicts with women's rights. The amendment merely states that citizens' tax dollars should not be spent in funding women's abortions. Therefore, if a woman wants to use abortion as a means of contraception, she can pay for it herself. If it's medically necessary, and she has insurance, the insurance should cover it. If not, she and her partner should pay financially for their decision.