The rise appears to be closely related to a rise in the civilian suicide rates, and even the Army, with the highest per 100,000 rate, is slightly lower than civil rate once the statistics are age adjusted. The adjustment is necessary as the civilian rate includes age groups far less likely to commit suicide, those groups not being well represented in the military (the military has a low average age when compared to the rest of society.) The percentages are too close to indicate the military is much safer or at greater risk.
Civilian rates have gone up and down over the years. The Army rates haven't been tracked as long, but appear to match the recent trend upward.
If there's a tragedy for the military, it is how the estimates suggest 65-70% of suicides are relationship related. The "Dear John" problem seems to be taking a heavier toll on soldiers than in the past. It's difficult to prepare a soldier for the possibility since they assume it won't happen to them, right up to the time when it finally does, at which point they aren't prepared to cope with the loss, and unlike their civilian counterpart, aren't there to argue the point face-to-face - aside from the fact that it's their absence which is often the cause of the breakup.
I'm basing this on the young male soldier as the stats for females seems a bit harder to check patterns - women are far more likely to attempt suicide, yet men are more likely to succeed as they tend to use instantly fatal methods.