You should check the cable for any markings. Usually you can then find the type by Google. Additionally, your switch should show which kind of SFP is inserted, regardless of DOM.
What's the approximate length of the cable?
Multi-mode (orange patch cable) supports 1000BASE-SX GbE for a few hundred meters (FDDI: 220m, OM1: 275m, OM2+: 550m). From ten years ago it's probably OM2 or OM3.
For 10GBASE-SR the ranges reduce to FDDI: 26m, OM1: 33m, OM2: 82m, OM3: 300m, OM4: 400m. If you require more reach on FDDI to OM2 you could try 10GBASE-LX4 (550m) but it is rare and costly. These reaches are by standard, you can often go somewhat further, depending on the exact fiber quality.
For single-mode fiber (yellow patch cable), you're probably good to go. 10GBASE-LR has the same reach as 1000BASE-LX(10) and you can buy higher-power transceivers with more reach.
On HP switches, you can query the transceiver models by
show interface transceiver detail - 1000SX stands for 1000BASE-SX (multi-mode) and so on. For 1000BASE-LX transceivers (single-mode), those equipped with DOM may show the Tx and Rx powers which gives you some estimate to what the link attenuation is.