Drinks before dinner at L'Espalier
You can sit in the Salon, an anteroom lounge at L'Espalier with comfortable seating, for cocktails before dinner: they're well-made, pricey, and convenient. I don't care at all for M Bar, the meat-markety cocktail bar associated with Asana on the hotel's first floor. Sel de la Terre in the hotel has a decent, more casual bar.
Other places that are an easy walk away with decent to very good drinks include Bar 10 (Westin Copley), the Hubcap Lounge (the bar at Top of the Hub, on the top floor of the Prudential), Abe & Louie's and Morton's (two luxury steakhouses nearby on Boylston), City-Bar (Lenox Hotel), Pazzo and Sonsie (both on Newbury Street), and Lucca Back Bay and Brasserie Jo (on Huntington Ave).
There is a bar at the Mandarin. There is also a Sel de la Terre restaurant, which is owned by the same group that owns L'Espalier. Never been to the Bay Bay one, but the one on the waterfront has a nice bar and good cocktails. And, it's close. Also nearby is Abe and Louis (across the street).
kwhite18's suggestions are spot on.
There isn't a bar in L'Espalier per se, but there is a "saloon", an oversized waiting area with plush chairs. They serve a basic set of house cocktails, Champagne, and various Champagne-based drinks, but something about it turns me off. They don't have a full bar, just a small butler's pantry stocked sufficient to make their house drinks. It is nice if you wanted a glass of Champagne while waiting for the rest of your party, but I wouldn't arrive early explicitly for a pre-drink cocktail. They also turn tables at L'Espalier with enough buffer that if you show up early and ask to get a cocktail in the saloon, you'll likely be given the offer to just take your seat early and enjoy a drink there. I like pre-dinner drinks to add another element to the evening, so I'd go elsewhere.
I'd actually recommend M Bar, the bar in the Mandarin Oriental, connected to Asana, their restaurant. Not because the bar is great--it is just okay--but it is so different from L'Espalier, it is a nice contrast, mixing up your evening. The bar in Sel de la Terre is good, but it is basically L'Espalier lite, and not a particularly interesting place. At least M Bar is a scene, and has full-length glass overlooking the street. If you were asking your question in a vacuum, without L'Espalier your ultimate stop, I wouldn't enthusiastically recommend it, but I think it fits in nicely with your evening. M Bar isn't going to craft you a crazy pre-prohibition cocktail no one has heard of, but they made me a fine Negroni last time I stopped in.
Another option is Oak Bar, adjacent to the Oak Room, in the Fairmont Copley Plaza, three blocks east in Copley Square. Solid cocktails.