Your question is nice, it shows that you have entered into running positively and possess a natural evolving ambition to improve.
I can give advice and suggest a regular schedule for you to build up your speed, keep in mind that you can fine tune my advice and schedule so that it fits you.
Over the past six months you have gradually built up your own individual level of endurance and you want to step further, it's time for you to set some regular weekly runs which require you to up your pace, run and run hard and don't forget to take out a day for rest when you do a really challenging run. Before running get in 10 minutes warm up time and after a run put together a stretch regime which works for you.
Check out this schedule, use it as it is or fine tune it to fit your availability.
Prepare yourself psychologically because you are going to train hard, you could imagine your training up for a big race competition and you want to be the winner.
This is a weekly training plan, roll it over each week and adjust your running activities to match your increased endurance, by week two or three you can adjust it to suit you or you will feel ready to take a 5k run and test our your endurance and speed.
Also give attention to your balanced diet, protein and energy food, fruits, vegetables, keep your body hydrated and make sure you sleep well and get plenty of rest.
Day 1. The start of your training schedule. fast but comfortable pace, run 1 k three times with around three minutes rest between each run.
Day 2 or 3 if you want day 2 for recovery. Take an easy run, this is about recovery from day 1, just tune in and focus on your technique, cadence, breathing, balance, run at a pace which would let you chat as you run, enjoy yourself and go for around 40 or up to 70 minutes, you can choose.
Day 4. Follow this closely, it's important. Think of your target speed and run for 5 minutes just a little slower than your target pace. Then, drop your pace for three minutes, look at yourself as a car cutting back a bit on the accelerator, not to slow, just drop your speed a little. Now take a 4-5 minute break, do this 4-5 times times, you choose but remember to challenge yourself, don't be tempted to under train.
Day 5. Take a day for recovery and adaption, put in a regime of stretch activities, your doing this for increasing your motion ranges.
Day 6. Speed training, going for a pace which is a little faster than your target speed, run for one minute, really put on the heat but stay within a realistic ability. Do these one minute speed runs eight times. Take out two minutes recovery time between each run. Keep going for maximum but realistic speed each time.
Day 7. Take an endurance run for about sixty minutes, push yourself, focus on your cadence, let your body work for you.
Choose a regular rest and recovery day, try out some regular stretch routines.
Remember as you work through this basic schedule, you can adjust your training pace, you will feel and know when your speed target has been reached.