B737 pilot reported a #1 Engine Hydraulic Pump failure; resulting in an inflight diversion to a suitable airport.

Narrative

Ferry flight (empty plane) from ZZZ to ZZZ1. Previous logbook write up showed left engine driven hydraulic pump low pressure" light came on during taxi out; pump replaced. On climb out; ENG 1 HYD PUMP LOW PRESSURE light illuminated. QRH non normal checklist "Hydraulic Pump LOW PRESSURE" was followed and pump was turned off. SYS page was brought up on the MFD (Multi-function Flight Display) to monitor hydraulic pressure and quantity. Noticed quantity A % was high at 106 during climb but good pressure. We reviewed other hydraulic non normal checklists for preparation of any further issues. After level off we noticed quantity A level was stabilized in the mid 80s (same amount as preflight on the ground). We discussed divert or precautionary divert possibilities if situation worsened. A short time after stabilized quantity (maybe 10 mins) we started to notice the Quantity A % slowly decreasing. I believe it was at this time we elected to precautionary divert to ZZZ2. As we were setting this up (notifying ATC/company etc) and heading that direction we noticed the quantity decreasing much more rapidly down to the mid teens in Hydraulic quantity A %. We decided to request priority handling and divert to ZZZ3; which we were directly overhead. The divert; descent; configuring and landing were all fairly eventful. We had good gear indications and good pressure indications even though low/depleted quantity indications. We were instructed to stop on the runway and CFR (Crash Fire Rescue) notified us that there was a large puddle of what appeared to be hydraulic fluid under the #1 engine. We were towed to a hard stand. When we deplaned a mechanic had the cowling open and said there was hydraulic fluid everywhere with a long trail of it."