Short answer: No. There are numerous effects which can enhance the eccentricity of BH binaries. This is a very active area of study (and thus fairly uncertain), but the leading mechanisms are tangential-deceleration preferentially at apocenter (by, e.g. dynamical friction or [perhaps] stellar scattering in massive BH systems) or excitation from a perturber (almost exclusively a stellar-mass BH-binary, with a massive tertiary). The problem is that these effects act slowly, on secular timescales, while the Gravitational-Wave (GW) damping of eccentricity is extremely effective as the system nears inspiral --- and acts rapidly, on a dynamical timescale.
To observe the effects of eccentricity during merger, you would basically need to excite a near-unity eccentricity in only the final orbit --- which is extremely difficult to do (cough, effectively impossible, cough).
Just to be an especial-downer, let me point out that this doesn't really matter because high-eccentricity effects are practically unobservable. This is because LIGO detections are based on templated searches which rely on observing hundreds to thousands of oscillations to make a statistical detection. To observe high-eccentricity effects at the time of inspiral would require really high signal to noise. We can make a quick estimate by noting that the Signal-to-Noise Ratio (SNR) is proportional to $1/r$ where $r$ is the distance to the binary (note this is not $1/r^2$). A binary blackhole can be detected at an SNR of about 8 in each of 3 detectors, at about 1 Gpc. Lets say that a factor of 10 higher SNR would yield observable eccentric effects (I think it's probably a lot worse than this...), which would require a 10x closer system, sampling a $10^3$ smaller volume of space, and thus a $10^3$ lower detection rate. Let's say the first binary BH detection occurs after a month of operation (and we haven't heard yet...), this would suggest it would take another 100 yrs before a sufficiently nearby system were observed. If we're lucky, LISA might be up within roughly that time --- which would have much better chances!