Unless your local area is unusual, you're unlikely to get a renewal in the timescale you want. In this area, the renewal of a temporary visa (or change to permanent) has to be applied for during the 30 days before expiry (not before). Some areas it's even nearer to the expiry date. So during early June you could get the application in. How soon it starts being processed will vary (can be weeks, and usually after the expiry date).
At some point later you'll need to be present for the fingerprinting, and another signature. This is very likely to be after your current expiry date (and after you leave the country). These are things an agent can't do for you.
Then the card needs to be processed, received, you notified of it, and you pick it up in person (I've heard this needs to be within 60 days of issue).
In this area, a good timescale for a change from Temporary to Permanent (after 4 years on Temporary and previous FM2/3) is around 10-12 weeks. Some are nearer 6 months. A renewal of a temporary is often faster. The temporary residence visa available for 1-4 years, in most areas, so it's best to get the maximum time and avoid annual renewals, where possible.
It's hard to see any way to speed it up, you probably can't apply earlier and realistically it probably won't start to be processed until around or after the expiry date (you can apply for a letter to be able to leave the country and return while the application is in process, there is a fee).
It could be well worth changing your travel plans (or at least making them flexible), as if you're grandfathered in on the existing rules you have big advantages over somebody starting from scratch from outside the country. Don't let the existing visa lapse, make sure you get the renewal/upgrade in, and be prepared to return at short notice to keep things moving along.
A good agent in your area might help (if they keep on top of things) but there are still going to be things you need to be personally present for, within fixed time limits.