There is no way to add additional ports to the switch you currently have; it is a "fixed" switch meaning that there's nothing you can do to expand it, etc.
You could connect a second switch to the first switch, though you'll have a bottle neck at the uplink from the new switch where it connects to this one... meaning that all devices on switch 2 will share the bandwidth of the single uplink connection when talking to anything on switch 1 or out through the router (presumable to the Internet). While this may not prove to be a problem, it is something to take into consideration.
The best option, however, would be to replace the switch with a larger single switch; ideally one that is expandable (through stacking or higher bandwidth uplinks).
Given the switch model you currently have, if I were in your position, I'd probably replace the current switch with a 48-Port switch from either the 3850, 3650, or 2960 series. Some options are:
3850: WS-C3850-12X48U - This 3850 offers 48 ports, of which 12 offer MGIG (100Mbps / 1 / 2.5 / 5 / 10-Gig) capability (the remaining 36 are standard 10/100/1000). It is equipped with an 1100W Power Supply, and a second one can be added for redundancy. The stacking technology on this switch is Cisco's tried and true StackWise-480 and StackPower is also offered. This is a layer 3 switch which means it can be setup to do routing between VLANs if necessary and is probably the best (stackable -- non-chassis) access switch on the market today. Also, an expansion module can be added to this switch providing 1-Gig, 10-Gig, or 40-Gig uplink ports which can be connected to an upstream distribution switch.
3650: WS-C3650-12X48FD - This is very similar to the above mentioned 3850, but with no stacking capability. If you should desire uplink ports in the future, that will need to be determined prior to ordering this model as a slightly different model number will need to be acquired depending on what uplink type you want (4 x 1-Gig, 2 / 4 / 8 x 10-Gig, or 2 x 40-Gig).
2960: WS-C2960X-48FPS-L - This 2960 has 48 10/100/1000 ports plus 4 SFP ports which can be used as uplinks in the future. It offers optional stacking and a 740 Watt POE budget.
A comparison chart is available between these three series of switches here: http://www.cisco.com/c/dam/en/us/products/collateral/switches/catalyst-2960-x-series-switches/sales-tool-c96-730473.pdf
As a personal note, I am a huge fan of the 3850's and have been using them since they were introduced. I also had great experience with the predecessor series to the 3850's (the 3750, 3750G, 3750E, and 3750X Series of switches). The stacking technology is very well tested and documented and has been in use for many years now (and has been upgraded in bandwidth over time since the 3750 / 3750G / 3750E / 3750X series).
One last option to mention is the brand new Cisco Catalyst 9000 Series. Something like the C9300-48U-E would give you all the functionality you have now, plus UPoE which means you could power any newly available devices which are introduced requiring more PoE power. These switches are brand new (announced within the last month) and are really exciting from a design / architecture perspective as they will lead us into the future of SD Networking. It might be another route to consider.
