Start where Beagles actually leave
Beagles rarely leave home by dramatic leaps. They leave nose-first, at ground level, through the gap that was almost too small, under the fence where the soil was soft, or through the gate that was almost latched, and by the time anyone notices, the trail is more interesting than your voice. Escape-proofing therefore starts with a ground-level audit of the whole property. Crawl the boundary at Beagle height with a torch, inside and out: look for dips beneath panels, gaps at gate posts, loose boards behind shrubs and any spot where the soil is dark, soft and already scratched. A Beagle reads those places as doors with the paint worn off, and your job is to repaint every one of them shut.
Under: hardening the fence base
Digging is the Beagle's signature exit, so give the base of the fence a proper defence. Bury mesh along the inside of the line, angled into your garden, or lay a run of paving slabs tight against the panels where the ground is soft, concentrating on corners, gates and any stretch where wildlife passes on the far side. Fill and firm any existing scrapes rather than just kicking soil back, because loose fill is an invitation to resume the project. If your Beagle has a favourite digging spot, make it legal instead: a designated digging patch or sandpit in the garden, stocked with buried toys, gives the behaviour an address that is not your boundary, and many Beagles accept the trade happily once it pays.
Through: gates, doors and the human factor
More Beagles leave through gates and doors than through fences, and almost always with human help. Fit gates with latches that close positively and cannot be lifted by a nose, add self-closing hinges, and teach the whole household the pull-and-check habit, calling out when the gate is shut. At the front door, install the airlock principle: a baby gate, a porch door or a closed inner door between the Beagle and the street, so a delivery never opens a straight line out. Put him behind that barrier before parcels, visitors and children come and go, every time, without exceptions for good behaviour, because the escape you prevent is the one that happens on the day he is not behaving but hunting.
The nose: management, not just barriers
Barriers buy time; occupation decides whether he spends that time testing them. A Beagle whose scent drive is fed daily, with tracking games, scatter feeding, scent trails laid around the garden and proper walks where he is allowed to sniff thoroughly, is a dog with less reason to prospect your fence line. Keep garden time supervised while you learn his habits, bring him in when neighbouring cats, foxes or rabbits are active at dawn and dusk, and store outdoor attractions, such as bins, bird food and compost, so their scent does not concentrate along the boundary. Check his collar fit and identification regularly, since a Beagle who does get out travels, and he travels at exactly the height where long grass hides small dogs.
If your Beagle follows a trail beyond your proofing, he may go a long way with his nose down before he notices he is lost, so KinsLe's Lost Pet Alert System helps by alerting nearby members at once and letting a finder message you directly through his profile.